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IN  MEMORIAM 
FLORIAN  CAJORl 


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in  2008  with  funding  from 

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http://www.archive.org/details/exercisesinalgebOOrobbrich 


EXERCISES  IN  ALGEBHA 


BY 

EDWARD   R.    ROBBINS 

M 

AND 

FREDERICK   H.    SOMERVILLE 

WILLIAM  PENN   CHARTER  SCHOOL,    PHILADELPHIA 


NEW  YORK  •:•  CINCINNATI  •:•  CHICAGO 

AMERICAN    BOOK    COMPANY 


Copyright,  1904,  by 
EDWARD  R.   BOBBINS  and  FREDERICK  H.   SOMERVILLE. 

Enteeed  at  Stationers'  Hall,  London. 


K.   A   8.    EXERCISES   IN   ALG. 
W.   P.    I 


Tit. 


PEEFACE 

The  present-day  teacher  of  Algebra  has  little  time  for  the 
selection,  from  proper  sources,  much  less  for  the  making,  of 
suitable  examples  often  necessary  to  supplement  those  con- 
tained in  the  standard  text-books.  This  little  book  is  designed 
to  meet  the  requirements  of  those  teachers  who  feel  such  extra 
assignments  in  Algebra  essential  to  thorough  familiarity  with 
its  processes.  The  aim  has  been  to  provide,  as  compactly  as 
possible,  a  series  of  exercises  that  conform  in  arrangement  with 
the  order  of  the  leading  text-books,  and  that  both  in  degree 
of  difficulty  and  in  scope  shall  include  the  work  prescribed  by 
high  schools  and  academies,  as  well  as  university  and  college 
entrance  requirements. 

The  plan  has  been  to  avoid  all  examples  of  more  than  aver- 
age difficulty,  and  to  lay  particular  stress  upon  those  subjects 
that  stand  so  clearly  as  the  foundations  of  later  work.  With 
this  in  view,  much  emphasis  has  been  given  to  the  subjects  of 
Factoring,  Fractions,  Exponents,  Equations,  and  Logarithms. 
There  has  been  constant  effort  to  present  abundant  drill  in 
topics  in  the  mastering  of  which  students  seem  to  have  the 
greatest  difficulties. 

Not  only  must  the  pupil  who  is  to  master  the  science  solve 
a  multitude  of  exercises  differing  in  degree  rather  than  in 

S 


4  PREFACE 

kind,  but  he  must  also  be  taught  to  select  appropriate  methods 
for  each  of  a  miscellaneous  collection,  differing  in  kind  rather 
than  in  degree.  To  this  end  the  book  is  generously  provided 
with  reviews,  and  the  pupil  is  obliged  to  discriminate  among 
different  kinds  of  examples  —  a  phase  of  mathematical  train- 
ing which  will  be  invaluable  to  him  in  subsequent  study. 
Furthermore,  the  reviews  are  so  arranged  that  they  could,  if 
desired,  be  used  independently  of  the  other  exercises  in  final 
review,  or  in  final  preparation  for  college  examinations. 

The  authors  will  welcome  and  will  appreciate  any  sugges- 
tions or  corrections  from  other  teachers  of  Algebra. 

EDWARD   R.   ROBBINS. 
FREDERICK   H.   SOMERVILLE. 

Philadelphia,  Pa. 


CONTENTS 


PAOK 

Substitution 7 

Addition 9 

Subti'action 10 

Use  of  the  Parenthesis 13 

Review 16 

Multiplication 17 

Division 18 

Multiplication  by  Inspection 20 

Division  by  Inspection 23 

Use  of  the  Parenthesis  with  Multiplication 25 

Simple  Equations 26 

Problems  in  Simple  Equations 27 

Review 29 

Factoring 31 

Review  . 38 

Highest  Common  Factor  and  Lowest  Common  Multiple     .         .  40 
Fractions  : 

Transformations 42 

Addition  and  Subtraction .45 

Multiplication  and  Division 48 

Complex  Fractions 50 

Fractional  Equations : 

Numerical  Equations 55 

Literal  Equations 59 

Simultaneous  Equations : 

Numerical  Equations 61 

Literal  Equations 64 

Three  or  More  Unknown  Quantities 66 

Problems  in  Simultaneous  Equations 68 

6 


6  CONTENTS 


Involution  and  Evolution : 

Monomials 71 

Involution  —  Binomials 72 

Evolution  —  Square  Root .72 

Evolution  —  Cube  Root 74 

Evolution  —  Numerical .        .        .        .        .        .        .        .74 

Review 7G 

Exponents : 

Transformations 81 

Miscellaneous  Applications    . 88 

Radicals : 

Transformations .        .         .93 

Miscellaneous  Applications 99- 

Imaginaries 103 

General  Review         .        . 107 

Quadratic  Equations : 

Numerical  Quadratics 116 

Literal  Quadratics 119 

Equations  in  Quadratic  Form 121 

Simultaneous  Quadratics 123 

Properties  of  Quadratics 126 

Ratio  and  Proportion .        .        .  129 

Variation 132 

Arithmetical  Progression 134 

Geometrical  Progression 138 

Permutations  and  Combinations 142 

Binomial  Theorem 145 

Logarithms 148 

General  Review 156 


SUBSTITUTION 

Exercise  1 

Find  the  numerical  value  of  the  following : 
When  a  =  1,  6  =  2,  c  =  3,  d  =  4. 

1.  a  +  b.  10.  7b-(c  +  d), 

2.  a  +  &  +  c.  11.  a  +  ab. 

3.  a-f26  +  3c.  12.  3a6-c. 

4.  a  +  36  — d  13.  12a  — 3c  +  cd 

5.  2  a  4- 4 6  — 2d  14.  3a  +  &(a  +  c). 

6.  6a  — 6  — d  15.  ab-}-a{2b  —  a). 

7.  10c-56  +  2d  16.  4a6c-3(c  +  d). 

8.  3a  +  c  +  d  17.  a(a  +  b  +  c). 

9.  3a4-(c4-d)-  18.  a6(a  +  6  +  c). 

19.  a6(d-a)4-&c(c?-6). 

20.  a  +  ab(b  +  c)-c(3d-3c), 

21.  c2-|-a6.  25.  ab^d  -  a(b^ -^  c). 

22.  62  4.c2  +  cZ2.  26.  25d-a62(a  +  6  +  c). 

23.  2a'b^  +  Sb^-G'  +  d.  27.  a  +  (a  +  6)1 

24.  &V  +  6c2  +  d  28.  4  6cd  +  (2  6  +  c)2. 

29.  3(a  +  2)2-2(62_l)  +  3a26c2. 

30.  2(a-{-by-S(d-by-bc{c-hd). 


When  a  =  5,  b  =4:, 

31.  V6  +  Ve. 

32.  Vie— V2d. 


SUBSTITUTION 
c  =  S,  d  =  2,  e  =  9. 


33. 
34. 

V5  ab  +  V4  6. 
V3  ce  —  V3  6c. 

35. 

•y/abc  4-  6. 

37. 

Va^  4-  6'  +  8. 

38. 

V2c2  +  3d2-a. 

39. 

a  +  b  -Vbcd  -f  1. 

40. 

(a  +  6)V6cd  +  l. 

36.    -Vabc  —  Sc  —  d. 


41.  (ad  —  e)  V  3  ac  —  e. 

42.  (a  + Ve)-(e- V6). 


When  a  =  4,  6  = 

=  5, 

m 

=  6, 

n  =  10. 

43.    ^« 

m 

19     3a  +  26 
n  — 1 

44     ^  +  ^ 

2a 

3  a  +  (w  —  -wi) 
*        2(n-a) 

45.    ^(^^-^). 
a 

51.    ^4-^  +  ^. 
m     n      0 

46.    ^(^  +  ^). 
a6 


b  n  2a 


47. 


m-\-n  —  g 
m 


/  V2a6  +  am\         , 

oo.    I ; ]-T  ^n  —  m. 

\      m  +  n      J 


48.    ^(^-^). 

Whena;  =  i   2/  =  |, 

55.  iC+2/  +  ^- 

56.  2x  —  y  —  z. 

57.  a;(x  +  ?/). 

58.  2/ (2  a;  — 2;). 


54. 


^m 


^n—m     Via 


+  V3 


2;  = 


59.  {x-[-y){y-z). 

60.  aJ2/(a;+?/-2;). 

61.  {x  +  lf-{y^-iyH^  +  iy. 


62.    aJ  +  2/ 


("J- 


(0^  +  ^)^. 


ADDITION  y 

When  a  =  4,  6=5,  m  =  2,  w  =  3. 

63.  or.  68.    (a  +  6)"  —  (a  +  6)"*. 

64.  a'^  +  6".  69.    (26-a)~-(36-a)'". 

65.  a^'  —  b'^.  70.    (a +  6  —  7)"'. 

66.  2a''  +  3a"'b\  71.    (a'^  +  ft"')". 

67.  (a  +  6)"'.  72.    a~  +  a'"(6  -  a)". 

ADDITION 
Ii2:ercise  2 

1.  Find  the  sum  ofa  +  36  +  c,  2a4-76  +  2c,  and  3  a  +  2  6 

+  c. 

2.  Find  the  sum  of  4a  +  36  — c,  2a4-264-4c,  and  a  — 
3  6 -2  c. 

3.  Findthesumof  10a-36-2c-d,  -2a  +  4:b-\-c  +  Sd, 
and  c  —  3  d  +  a. 

4.  Add  3a  +  26-3c,  12a-46-7c,  and  4a-86  +  9c. 

5.  Add  3a^  +  2a^-2«  +  ll,  4.a^-2x^-\-3x-S,  4.a^-2x' 
-\-x-2,  -12ar^  +  a;2-x-l,  and  2 o^ -\- oF -  x -\- IS. 

6.  Collect  5a-3c  +  4/-m  +  26-d  +  4c-2a-3/+c 

+  2  m  +  d. 

7.  Collect  2  a6  4-  3  6c  +  4  cd  —  2  6c  +  3  a6  —  3  cd  —  4  a6  - 
2  6c-2cd 

8.  Collect  2a6cd  +  3  6fljH-m  — 3a6cc2  4-2  6a;  — 3m  +  6a6cd 
—  3  6»  +  3  m. 

9.  Collect  a^-hSa^  +  Sa,  a^  +  a  +  l,  and  2a  +  2. 

10.    Collect  a^  +  a'b-\-ab%  3a^  +  a'b^-2b%  and  4a62-3  6^ 


10  SUBTRACTION 

11.  Arrange  in  descending  powers  of  x  and  collect  a:^  —  2  + 
^x'-x,  -a;4-3a^-2ar*,  _4ar'  +  5a^-a;4- 10,  -a;-f4a^- 
2  +  ic8,  and  a^  +  a;^  _,_  ^  _,_  ^^ 

12.  Arrange  in  ascending  powers  of  a  and  add  a^  —  4  a^4- 
2a-8,  a^  +  a-3a2  +  16,  a34-a'-2  +  3a,  and  -^.a'  +  Qa? 
+  10  a  -  3. 

13.  Find  the  sum  ofa^-l+3a;-a^  +  2a;-3a:3  +  44-«^- 
8a;  +  2a.'^  —  4a^  +  10  —  ic^,  and  arrange  the  answer  in  descend- 
ing powers  of  x. 

14.  Add  3(a4-6)4-4(c  +  d)-5(w4-n),  2(a  +  &)-2(c  +  c?) 
-\-Q{m-\-  n),  and  —  4  (a  +  6)  —  (c  +  d)  —  (m  +  n). 


15. 

16. 

17. 

18. 

O/C 

ac 

mx 

my 

be 

c 

5x 

-Sy 

{a-\-b)c  (a  +  l)c  (         )»  (         )y 

19.  Add  ab  +  cc?  and  mb  +  ncZ. 

20.  Add  ax  +  by  and  ca;  +  dy. 

21.  Add  2  a6  +  3  c  4-  c?  and  3  6  +  oc  +  nd 

22.  Add  3a6+2ac+w,  36— 3ac+mw,  and  2 nib +4: be— pn. 

23.  Add  (a  +  &)c  +  (6  4-  (^)c. 

24.  Add  (m  +  7i) a;  +  (m  —  n) a;  —  mx. 

25.  Add  (2  +  a)a^  +  (3a-4)a^+(a-l)ar^-4aar^. 


Subtract : 

1.            2.  3. 

10  a        10  a  -  4  a 

3  a      —4  a  3  a 


SUBTRACTION 

Exercise  3 

4.              5. 
-3  a       15a' 
-a      -2a' 

6. 

4a6 
3a6 

7. 

3a6 
7a6 

8. 

2  am 
—6  am 

SUBTRACTIOn 

11 

9. 

10.                    11. 

12. 

13. 

3aH-26 

14a  +  3c      16a-2 

c       12  aft  +  16 

—  5a  —  4  c 

a+     h 

7a—     c        3a  +  5 

c        3  a6  -  10 

-2a-5c 

14. 

15. 

16. 

17. 

a^  -  10  ah 

2  a^  +  12  a?7i 

5a6 

16  am  4-  mn 

a^H-    3a6 

3a2+    3  am 

4a6  +  cd 

Sam 

18. 

19. 

20. 

4  a^c  4-  3  a6  +  10           15  a? 

10  a^  +  3 

!  a^  4-  6  a  +  7 

-2  ah 

10a2  +  4a 

+  4 

a^            -4 

21.  Take  a^h  +  c  from  4a  +  364-2c. 

22.  Take  5  a  —  6  4-  c  from  6  a  +  3  6  —  4  c. 

23.  Take  2a4-36— 4c  from  4a4-36H-8c.      , 

24.  Take  3  a  + 10  6  -  14  c  from  4  a  4- 10  6  -  14  c. 

25.  Take  a-  +  3  a6  + 12  from  5  a^  -  8  a6  + 16. 

26.  Take  4a2  4- 3a6  4-2a&2_  10  from  8a2-16a6-3a62_l. 

27.  Take  3a2  4-3a  +  l  from  4a3  +  5a2-6a-3. 

28.  From  3  a^- 2  a^  +  e  a2  4-2  a-3  take  2a^-a3-4a2- 
a  + 1. 

29.  From  a* -  4 a^ft  +  6  a^h'  -  4  aW+h"^  take  a36+3  a%^+ah\ 

30.  From  -3a2  4-8a  +  36  take  a2-2a-18. 

31.  From  -16a3-8a2  4-4a-5  take   -2 a^ +  a'^ -a-\-l. 

32.  Given  a  minuend  a^4-4a-  — 3a4-2  and  a  subtrahend 
a^  4-  3  a^  —  4  a  4-  1,  find  the  difference. 

33.  Given  a  subtrahend  —  3  ac,  a  minuend  8  ac,  find  differ- 
ence. 


12  BUBTRACTION 

34.  Given  a  minuend  3  ahc,  a  difference  4  ahc,  find  subtra- 
hend. 

35.  Given  a  minuend  axy,  a  difference  —axy,  find  subtra- 
hend. 

36.  What  must  be  added  to  16  ac  to  make  —Sac? 

37.  What  must  be  added  to  75  m  to  make  31  m  ? 

38.  What  must  be  added  to  —  3  a^  to  make  0  ? 

39.  36  a^  is  added  to  what  expression  to  make  82  a^? 

40.  What  expression  added  to  4  a  -f-  2  c  will  give  5  a  +  8  c^  ? 

41 .  Given  a  subtrahend 4a'*4-3a^  —  2a^  +  a  —  7 and  a  differ- 
ence —  3  a^  +  2  a^  —  a,  find  the  minuend. 

42.  Given  a  minuend  8m^  — 3  mn  +  2  n^-f  18  and  a  differ- 
ence 2  m^  +  2  mn  —  n^-\- 16,  find  the  subtrahend. 

43.  Take  the  sum  of  4  m^  +  2  mn  —  r?  and  —  3  m^  —  mn 
-\-2n^  from  the  sum  of  3  m^  -f  12  mn  + 10  n^  and  —  2  m^  — 
11  mn  —  11  n^ 

44.  Subtract  x^-Soc^ -\-0(^ -2 —  3  x  from  a;  +  a^-3  +  2aj* 
—  X'  —  a?,  and  arrange  the  result  in  ascending  powers  of  x. 


45. 

46. 

47.                  48. 

49. 

50. 

ac 

am 

X                  2  ax 

5ahx 

am 

he 

m 

ex             —3  ex 

3cdx 

-2em 

(a-b)c 

(a  —  1)  m 

(l-e)x     (2a  +  3c)a; 

51. 

52. 

53.                       54 

55. 

3mn 

3aa;  +  2 

ax-\-by             x-\- 

y 

3 

ax-\-  y 

-2pn 

ca;-l 

ex  -\-dy            ex  — 

dy 

x-cy 

56.  Take  2  6c  — 3  ac?  from  3  ac  + 12  6c?. 

57.  From  ax -{- 3  ey -\- ^  dz  take  hx  —  2dy  —  az. 

58.  From  3 ax -\- 2 ay -\- 10  take  2x  —  y-\-l. 


USE  OF  THE  PARENTHESIS  13 

USE   OF   THE   PARENTHESIS 
Exercise  4 

Simplify:      1.  a +  (3 a +  2). 

2.  4a  +  (2a-10). 

3.  5a  +  (3a-2)  +  l. 

4.  2a;  +  (4aj-2/)  +  (3a;-2)+y. 

5.  (5m  +  7)4-(3m-2). 

6.  2c  +  (3cH-4)-(c  +  2). 

7.  5m-(m4-l)-(2m-l). 

8.  62/-(42/  +  32;)-(22/-42;). 

9.  3a  +  [a  +  (4aH-3)]. 

10.  8a  +  [a-(2a-l)]. 

11.  5  m  — J3mH-(2m-l)J. 

12.  2a  +  {-3  +  (2a-l)i. 

13.  lla-[4a-f  (lOa-6);. 

14.  -3a  +  S2a-(-a  +  l)S. 

15.  -2a-[-a-S-aH-lS]. 

16.  13a  +  l-4a-(10a-[a-l])J. 

17.  a+[a  — ;a— (a  — 1)J]. 

18.  a-[-a-5-a-(-a-l);]. 

19.  5  a  -  [2  a  -  (a  + 1)  -  {3  a  -  (a  + 1)  - 1 S  - 1]. 

20.  6  a  -  [(3  a  + 1)  4-  53  a  -  6  -  (a  +  2)  -  3  aj  -  a]. 

21.  (5a-l)-[5(3a-l)-10a  +  5J-a]-J-(a-2)j. 


22.    (3  -f  a  - 1)  -  [-  a  +  (a  -  2  a  -  3)  -  3  a]  -  {a  - 1  +  aj 


14  USE  OF  THE  PARENTHESIS 


23.  (4a2-[3a2-(a4-2)-4]-|a2_^3a-(a-l)|)-a. 

24.  l-(-l)  +  (-l)-|l-[l-(l_lTa)  +  a]-aj. 

25.  l_J-[_(-l)-l]-li-(-J-[-(-l)]J). 

26.  Sx-\a-{2a-l3a-{5a-[7a-{Sa-x)J)J)\. 

27.  -m-[Sn-{-\2p-Sm-(m-n)l-{-7i-\Sm-{m-{-n)\^. 

28.  (4:a-a  +  2)-\a-a-^{a-2)-2\-a-a-{-l. 

Simplify  and  find  numerical  values  of  the  following  when 

a=  5,  6  =  3,  c  =  —  1,  d  =  —  2,  and  x  =  0: 

29.  a  +  6  +  c  4-  d  +  a;. 

30.  2a-\-c-3b-d. 

31.  a  +  2c-(d  +  2)+2(c  +  l). 

32.  a  +  c  —  d  — 36  + a;. 

33.  4  6  —  d  +  a6  —  aa;. 

34.  2a-(c  +  d)-(a  +  d). 

35.  4a6a;+(a  — d)  — (6  — c). 

36.  c +3a;  — 2(a  +  2a;)+4a. 

37.  a  +  2  6  — [3a  +  c  — S2a  — (a;  — c)|]. 

38.  a  +  3c  —  \b  +  2x—(a  —  c)l-]-2d. 


39.  3  a  — [5  6  — (a6  +  d)  — a-d]  — (a6  +  a;). 

40.  a2-5a;-[3a6  +  462-(a6a;  +  d)]. 

41.  aa;-[-64-(a-d)2  +  a]-26. 

42.  (a  +  c)2+(a-c)2-[a6  +  (2  6  +  a;)2]. 

43.  (a-\-c)x  —  (abx  +  a)^  —  (d  +  c). 


44.  (a  +  a;)  6  -  [  V2  a  +  c  -  (d  +  c)]. 

45.  ic-[c+(a  +  d)2-26]2+(a-V^6^^)l 

46.  a62-S3(6  +  c)2-[-c+2a(4d+6^]-[6ca;+V3a-c]j. 


I 


USE  OF  THE  PARENTHESIS  16 

Exercise  5 

1.  Insert  the  last  two  terms  of  a  +  b  —  c  in  a  parenthesis 
preceded  by  a  plus  sign. 

2.  Insert  the  last  two  terms  of  a  — 6  — c  in  a  parenthesis 
preceded  by  a  minus  sign. 

Insert  the  last  three  terms  of  each  of  the  following  expres- 
sions in  a  parenthesis  preceded  by  a  minus  sign : 

S.    a  —  b  —  c  —  d.  e.   a-\-b  +  c-{-d. 

5.   a-^b-c-d.  8.   4a*- a^-f- a^- 3a  +  l. 

9.   6a^-a^-10a*-{-a^-a\ 

Collect  the  coefficients  of  a,  of  b,  and  of  c,  in  each  of  the 
following : 

10.  Sa  —  2b-\-c  —  ma  —  nb  —  pc. 

11.  5a  —  Smb-\-2nc  —  2a  —  6nb—3c  —  ma, 

12.  10a  —  4^b  —  2c  —  ma  —  nb-{-pc. 

13.  —Sa— 4:C  — pb  —  na  —  mc  —  b. 

14.  —c  —  a  +  4:b  —  Sc-\-da  —  10b, 

15.  —4:a  —  3mb  —  10c—b  —  ma  —  nb-\-pc^da. 
Collect  the  coefficients  of  like  terms  in  each  of  the  following : 

16.  4ar^-2ic  +  3a^-3a;  +  4a^-a^. 

17.  -Sx^-2x-\-x*-3x^  +  2x-x\ 

18.  aa^  —  ba^—cx  —  cx^  —  dx  +  a^. 

19.  a^  +  a^  —  aa;  —  ca^  +  3  a^  —  6a;. 

20.  —  aoi^  —  cc^  —  x  —  4:  x^  —  ax^  —  px  —  ca:^. 

21.  —px^  —  cx-\-3xr  —  mx  —  nx^  —  mx^  —  abx  +  Sdx^  —  cu?. 


REVIEW 
Exercise  6 

Find  the  numerical  value  of : 

1.  (x-\-2y-{x-\-l)-4:  when  x  =  S. 

2.  (x-2)(x-j-5)-2{x  +  iy  when  x  =  4:. 

3.  (a  +  6)2  -  2  (a  -  1)  (6  -  1)  -  ab-  when  a  =  5  and  6  =  3. 

4.  Vic^  +  m^  —  (n  —  myY  when  a;  =  4,  y  =  0,  m  =  3,  n  =  2. 

5.  2  a  —  [a  —  (3  a  —  2  6)]  when  a  =  3  and  6  =  2. 


6.  3a—  [4  6  +  2m  — 3c  +  (a  —  6)  —  2a]  when  a  =  4,  6  =  3, 
c  =  —  2,  and  m  =  1. 

7.  Subtract  the  sum  of  a^  +  a^  — a  and  2o?-\-2a  —  l  from 
the  sum  of  3  a^  —  a^  —  a  +  1  and  2  a^  +  4  a  —  3. 

8.  Take   m^— 3m^   from   the   sum.   of  2m*— m^  +  2   and 
m^  —  3  m  + 1. 

9.  From  the  sum  of  a^-{-a^  —  a  and  a^  +  4  take  the  sum 
of  4a3-2a2  +  17  and  a  -  3. 

10.  If  ^.  =  0^  +  0^4-1,  -B  =  2a;2-.T  +  2,  and  (7=a^-3a;  +  6, 
find  the  value  of  ^  +  J5  -  C. 

11.  With  values  for  A,  B,  and  C  equal  to  those  in  Ex- 
ample 10,  find  the  value  oi  A  —  B-\-  C. 

12.  To  what  expression  must  you  add  5  a^  to  make  0  ? 

13.  What  expression  added  to  a;^  +  a;  —  1  will  make  0  ? 

14.  —  cc^  —  a;  +  3  is  subtracted  from  2  a;-  —  3  a;  +  4,  and  the 
remainder  is  added  to  —  3a?2  +  2a;  — 14.  What  expression 
results  ? 

16 


MULTIPLICATION  17 

MULTIPLICATION 

Exercise  7 

Multiply : 

1.  a  +  1  by  a-f2.  6.  a2  + a  +  1  by  a  +  L 

2.  a  — 2  by  a  — 5.  7.  a- +  a  +  1  by  a  —  1. 

3.  a  +  4bya-3.  8.  a'^+Sa^+Sa+l  by  a+L 

4.  a- -hi  by  a2_^2.  9.  x'-2x^-{-l  by  a^  +  L 

5.  a--{-3  by  2a--5.  10.  4 a;' +  4 a.- +  1  by  2a;  +  l. 

11 .  a-  —  3  a  —  2  by  a-  —  a  +  1. 

12.  a''-Sa'--\-3a-l  hy  cr--2a-\-l, 

13.  12?/2-G?/-f-2  by  /-32/-f-L 

14.  a4_4^.3_|_(5^^2_4^^^  j^  ^y  a^-Sa^  +  Sa-L 

15.  5a^-2a2  +  3a-l  by  3a2  +  a-l. 

16.*  m^-2m''-h3m2-2m  +  l  by  m'^+2m^+Sm^-\-2m-^l. 

17.  m"  -  nr  -\-Sm  —  5  by  m^  +  m^  -f  3  m  +  5. 

18.  a-  +  Z>--2a6-46-h4«-f-4  by  a-6+2. 

19.  x^  +  x'^  +  l-\-x  by  1  +  a;2  4-  a;. 

20.  l-Yx'  +  x-'-^Bx  by   _  4a^+ 1 -f  2a;2. 

21.  l-4a'-4.a  +  Ga'-{-a'  by  Sa-l+a^-Sa^. 

22.  .t^-2.t2  +  3  by  l-aj2^if. 

23.  or  -\-2ab  +  h" -{-  7n"  by  a^  +  2a&  +  6^  _  -^2^ 

24.  a^-a^-^a-1  by  1  -f-  a  +  a^  4.  ^3. 

25.  5x'-2x'  +  x'-3  by  -  3x' -^2x' -2 +x\ 

26.  a^  H-  2  a5  -f  ?>^  by  ct"  -  3  a-&  +  3  ab'-  -  h\ 

27.  a^  -  4  o&-^  +  ^'  +  T)  a7>2  _  4  a-^&  by  52  ^  a-  -  2  a6. 

R.  &  S.  EX.  IN  ALG.  —  2 


18  DIVISION 

28.  3a^b-2  a^V  +  4  a6«  by  2  a^h  ^V-  ah\ 

29.  5  a^m  +  4  aW  —  3  «%^  —  am**  by  2  a^m  —  3  am^. 

30.  (a +  1)  (a +  2)  (a +  3). 
;a  +  2)(a-2)(aH-l). 
a  +  1)  (a  -  l)(a  +  3)(a  -  3). 
>  +  2)  (»  +  2)  («  -  2)  (a;  -  2). 
V  +  m-\-l)  {m^  —  m  -f  1)  (m}  —  m 

;3  a,.2  _  5  ^  _l_  3)  (^  _  4)  (^  _  ^  ^  2), 


31. 
32. 
33. 
34. 
35. 
36. 


-1). 


[a  -\-h)-\-2  by  (a  +  &)  +  3. 

38.  (a  +  2)  +  3  a;  by  (a  +  2)  +  4  a;. 

39.  (a  +  c)  —  4  by  (a  +  c)  +  6. 

40.  (a  +  6)  +  (c  +  d)  by  (a  + 6) +2(c  +  d). 


Divide : 

1.  a^  +  4a  by  a. 

2.  3a3  +  6a2  by  3a2. 

3.  a^  —  a^-\-or  by  a^. 

4.  a"*  — a^  +  a-  by  —a^ 


DIVISION 

Exercise  8 


5.  Sa  +  lOa^  +  loa'^  by  5a. 

6.  12m=^-9m2+3m  by  3  m. 

7.  —  ??i"^  4- ?yi- +  4  m  by  —m. 

8.  15a252_75a6  by   -5  aft. 


9.  27a^6^  +  36a26^-18a&^  by   -9a&^ 

10.  a^.^ 4a +  3  by  a 4- 1. 

11.  l  +  8a  +  15a2  by  l+3a. 

12.  a34-3a2  +  3a  +  l  by  a  +  1. 


DIVISION  19 

13.  Sa^-{-12a^-\-6a-{-l  by  2a  +  l. 

14.  16-32a  +  24:a^-Sa^  +  a'  by  2-a. 

15.  a^-4a3  +  6a2-4a  +  l  by  a'-2aH-l. 

16.  m^ -\- 5  m^n -{- 5  mn^ -i- n^  by  m-  +  4m7i  +  7t^. 

17.  4m^-9m2  +  6m-l  by  2m^-[-3m-l. 

18.  a^-10a*  +  40a3-80a2  +  80a-32  by  a2-4a  +  4. 

19.  m*-3m3-36m2-71m-21  by  m^-Sm-S. 

20.  4a^-15a-^c  +  26aV-23ac3  +  8c*  by  4a2-7ac  +  8c2. 

21.  m^-6m^  +  5m2-l  by  7/1^  +  2  7/i2_^,t_l. 

22.  a^  +  12a2-48  +  52a-17a3  by  a-2  +  a2. 

23.  m'*  4- 4  mV  +  16  71*  by  m^  +  2  mn -f  4  ?r. 

24.  18c*  +  82c2h-40-67c-45c3  by  5-4:C  +  Sc\ 

25.  4a;2-(-13a;-6a:3  +  6  +  a^-2a;*  by  1 -{-x' +  3x  +  Sx^. 

26.  25 m -  lOm^  + 15  + 14  m*- 41  m^  by  -5m  —  S-{-7m^' 

27.  -14c*d  +  12c^  +  10c3cZ2_c2d3_g^^^4_^4^^5  ^y 

2cZ3-3cd2-4c2d  +  6c^ 

28.  m^  —  n^—p^  —  2pn  by  m  —  n—p. 

29.  c^  +  d^  +  77i^— 3  cdm  by  c  +  cZ  4- m. 

30.  a'^4-6^  +  c3-3a6cby  a^  +  fts^c^-ac-ftc- a6. 

31.  x'-y^hj  x-y.  38.  64  +  27  a*^  by  44- 3  a. 

32.  o^-fhj  x-y.  39.  16-81  a*  by  2 +  3  a. 

33.  a;*-2/*by  a;-2/.  40.  125  0^-8  by  5  c- 2. 

34.  x*-y^hj  x  +  y.  41.  216  771^-27  by  6  m-3. 

35.  «3-27bya;-3.  42.  81  a;*- 16/ by  3a;  +  2  i/. 

36.  a;*-16by  a;  +  2.  43.  9m*-49  c«  by  3  7n,2^7  c*. 

37.  32-m^by2-m.  44.  27  d^  64  6^  by  3  c^H  4  6'. 


20  MULTIPLICATION  BY  INSPECTION 

MULTIPLICATION   BY   INSPECTION 
Exercise  9 
The  square  of  the  sum  of  two  quantities. 
Expand  by  inspection : 

1.  (a +  6)2.  5.    (a +  3)2.  9.    (3  a +  0)1 

2.  {a  +  cy.  6.    (a +  5)2.  lo.    (2a  +  3a.f. 

3.  (a  +  m)l  7.    (a +  10)1  11.    (J  ax-\-^y. 

4.  {a  +  2y.  8.    {2a^hy,  12.    {^a?  +  dhy. 

The  square  of  the  difference  of  two  quantities. 
Expand  by  inspection : 

13.  {a -by.  17.    (a-2cy.  21.    {2a-^cdy. 

14.  (a -my,  18.    (3  a -2)2.  22.    (3a2_2a5)2. 

15.  (a -4)2.  19.    {db  —  cy,  23.    {4.ax-xyy. 

16.  (a -6)2.  20.    {a?-ahy.  24.    (9a2-10c2)2, 

The  product  of  the  sum  and  difference  of  two  quantities. 

Multiply  by  inspection : 

25.  (a  +  6)(a-6).  30.  (2  a6  +  1)(2  a&  -  1). 

26.  (rt +  c)(a-c).  31.  (4a+3)(4a-3). 

27.  (a  +  2)(a-2).  32.  (pa^ -2c)(p  0^  +  2  c). 

28.  (a-4)(a  +  4).  33.  (S>  a" -7  ah)(Qa? +  1  ah). 

29.  (a2-3)(a2  +  3).  34.  (5a;^/-3a^)(5a;^/+3a;;^). 


MULTIPLICATION  BY  INSPECTION  21 

Perform  the  indicated  multiplications: 

35.  {a-2c)K  41.    (a6c-2)2. 

36.  (a-{-Sxy.  42.    {arm -\- mn){am  —  mvi). 

37.  (a2-2)(a2  +  2).  43.    (4  ac  +  7)(4ac-7). 

38.  (a^-l)(a^  +  l).  44.    (6m2-3)l 

39.  (a3  +  3)2.  45.    (8m-h5)(8m-5). 

40.  (a6  +  5c)2.  46.    (3  (r'  +  5)(3  c*- 5). 

The  product  of  the  sum  and  difference  of  two  quantities  obtained 
by  grouping  terms. 

Multiply  by  inspection : 

47.  [(a  +  ?>)+c][(a+6)-c].  55.  (a'+a-irVjipi^-^-a-l), 

48.  [(a4-«)+3][(a+a;)-3].  56.  {a-x+y){a-x-y). 

49.  [(a-2)+c][(a-2)-c].  57.  (a-x-^y){a-irx+y). 

50.  [(a2+l)+a][(a2+l)-a].  58.  (c-d4-3)(c+d+3). 

51.  [cH-(a+6)][c— (a+6)].  59.  (a+m— n)(a— m+zi). 

52.  [m4-(w-p)][m-(n-i))].  60.  (c+cZ-3)(c-d+3).      . 

53.  (a  +  6  +  c)(a  +  &-c).  61.  (x2-l+a;)(ar^-l-ic). 

54.  (a+a;+2/)(a+a;— 2/).  62.  (o^+a;— l)(a;2_^_j_j^)_ 

63.  [(a  +  &)  +  (c  +  ^)][(a  +  &)-(c  +  cr)]. 

64.  [(m  +  w)  — (a;-2/)][(m  +  n)  +  (a;-y)]. 

65.  (a  —  m  +  n  — l)(aH-m+n4-l). 

66.  [m«-3m2-m-3][m3-|-3m2  +  m-3]. 


22  MULTIPLICATION  BY  INSPECTION 

The  square  of  any  polynomial. 
Expand : 

67.  (a  +  5  +  c)2.  71.  (a^  +  a  -  2)1 

68.  (a  +  6-2c)2.  72.  (2x^ -S  xy  -  5f)K 

69.  (a-\-b-c-  df.  73.  (a^  +  2  a^  -  2  a  -  3)1 

70.  (a +  26 -3c +  («)'.  74.  (2a^-ar^2^+3a;2/'-22/3>)2^ 

The  product  of  the  forms  {x  ±a){x  ±b). 

75.  (a;  +  2)(a;  +  3).  82.  {x-^){x  +  12). 

76.  (a;  +  3)(a;  +  5).  '  83.  (a^- 3)(a^  +  7). 

77.  (a?  +  10)(a;  +  7).  84.  (a:^  _|_  4^)(^^  _  i3>)_ 

78.  (a;  +  3)(a;  -  2).  85.  (a^  +  2)(a.'3  +  19). 

79.  (a;  +  4)(a;-5).  86.  (y?f  ^-l){x'y'' -2). 

80.  (a;-4)(a;  +  l).  87.  (aa;  +  13) (aa;  -  12). 

81.  (a7  +  3)(a;-16).  88.  {a'cx-ll){a^cx -\-Z). 

The  product  of  the  forms  {ax  ±  b)(cx  ±  d). 

89.  (2  a;  +  1)(3  a;  +  5).  95.  (a' -  3  a)(4:  a^ -\- 2  a). 

90.  (3  a;  +  2)(4  x  +  3).  96.  (4  ac  + 7  m)(3ac- 67/1). 

91.  (3a-7)(2a  +  3).  97.  (llx'-3y%5x' -\-f-). 

92.  (3m-^Sx){Sm-llx).  98.  (mn  -  13  1/)  (5  mw  +  y). 

93.  (4a  +  3c)(3a  +  2c).  99.  (llm^-37i){3m^-\-nn). 

94.  (2c2-7)(3c2-ll).  100.  (6cd-3dm)(4:cd-\-Sclm), 


DIVISION  BY  INSPECTION  23 

DIVISION   BY   INSPECTION 
Exercise  10 
The  difference  of  two  squares. 
Divide  by  inspection : 

1.  ^2- 62^  a- 6.  6.   25m2-81-^5m^-9. 

2.  a^-9-^a-\-3,  7.    81  ?>i^  -  49 -- 9  m^  -  7. 

3.  ci2_16--a-4.  8.    64  a^  -  36  ^  8  «  +  6. 

4.  4a2_l^2a+l.  9.    169  a" -9 -r- 13  a +  3. 

5.  16a2-9-j-4a-3.  10.    4m^  -  225 -^  2m2- 15. 

The  difference  of  two  cubes. 
Divide  by  inspection : 

11.  m^-ji^-j-m-n.  17.  8  a«  -  1  ^  2  a  -  1. 

12.  a^-b^-i-a-b.  18.  125  m^  -  27 -- 5  wi  -  3. 

13.  m^-l-i-m-l.  19.  1  -  512  m«-^  1  -  8  m^. 

14.  C3-8--C-2.  20.  343(^-d^^7c-d^ 

15.  2T-x^-^3-x.  21.  125-216?7i»-r-5-6m3. 

16.  l-64a3--l-4a.  22.  1000  -  mht^ Sr  10  -  mil. 

The  sum  of  two  cubes. 

Divide  by  inspection : 

23.  m^ -\- 71^ -i-m  +  n,  26.    64a3  + 27  ^4  m  +  3. 

24.  8  +  C3-2  +  C.  27.    125a3  +  8--5a  +  2. 

25.  27m3  +  lH-3m4-l.  28.    a^m^  +  27  ^  am  +  3. 


24  DIVISION  BY  INSPECTION 

29.  a«m«  +  64 -r- aW  +  4.  31.    216  a^  +  512-f- 6  a +  8. 

30.  8a^-f343^2tt  +  7.  32.    1000 a^ + 729 ^10 a  +  9. 

The  sum  or  difference  of  like  powers. 

Divide  by  inspection : 

33.  a'^-b^-ha-b.  39.  a^  +  32 -j- a  +  2. 

34.  a^-b^^a-b.  40.  16a^-l-f-2a  +  l. 

35.  a^-b^-^a  +  b.  41.  32  a^ +  1-1-2  a +  1. 

36.  a^-\-b'-i-a-[-b.  42.  81  -  16  a* -J- 3  -  2  a. 

37.  a^  —  ¥-^a  —  b.  43.  64  —  m« -s- 2  +  m. 

38.  a^-16H-a-2.  44.  32  +  243  a^^  2  +  3  a. 

Give  the  binomial  divisors  possible  for  each  of  the  following : 

45.  a^-8.  48.    a* -81.       51.    9a*-16.      54.    a«-9. 

46.  a* -16.       49.    a^-64.       52.    81  -  a\       55.    a^-\-b^. 

47.  a^  +  32.        50.    a^-b\         53.    16  -  a^        56.    a^^  _  512 

Give  the  quotients  of  the  following : 

57.  a2-16--a  +  4.  62.    125  a^  -  8 -^  5  a- 2. 

58.  a^-16--a-2.  63.    100  c^  -  a^ -f- 10  c^  +  a^ 

59.  4a*-25--2a2+5.  64.   36  m*- 1  ^  6  m^-f  1. 

60.  81  a*  -  36  c^ -f- 9  a^- 6  c.       65.    64a«- 27c^2^4a2-3c^ 

61.  lOOm^-l-^lOm^-l.        66.    1  -  81  ««-^^  1  +  3  al 

67.  100(a4-l)'-9^10(a  +  l)  +  3. 

68.  27(a  +  l)^-8c-^--3(a  +  l)-2c. 


PARENTHESIS    WITH  MULTIPLICATION  25 

USE   OF   THE   PARENTHESIS   WITH 
MULTIPLICATIOK 

Exercise  11 

Simplify : 

1.  a4-(a  +  l).  9.  (a  +  4)(a-2). 

2.  a+(2a-3).  10.  2a-\-S(a-3). 

3.  a  +  2(a  +  l).  11.  {a-{-l)(a  +  2)-a\ 

4.  a  +  2(a-4).  12.  (a  +  2)^  +  (a  + 1)2. 

5.  a2+(a  +  l)'.  13.  (a  +  4)^  -  (a  +  2)2. 

6.  a  +  2(a-l).  14.  7  a^ -  2 (a^  +  1). 

7.  (a+2)(a-l).  15.  2(a4- 1)'- a(a  +  l). 

8.  5a  +  S(a  +  2).  16.  4(a  -  l)2-3(a  + 1)1 

17.  4a  +  (a  4- 1)  (a -1) -(«-!)'« 

18.  2(a  +  l)(a4-2)-(a  +  l)(a-2). 

19.  (a  +  l)2-(a  +  l)(a-l)-2a. 

20.,  3  a^  +  (a  +  3)  (a  -  1)  -  (a  -  3)  (a  +  1)  -  2  (a^  -  2  a). 

21.  2  (a  +  1)'  -  2  (a  +  1)  (a  -  1)  +3  (a  -  2)\ 

22.  (a  +  3)2-2(a  +  l)2  +  3(a  +  2)2. 

23.  (m  +  n)^  —  m{m-\-nf  —  n  (in  +  n)^. 

24.  m  (m  +  n)2  -|-  n  (m  —  ?i)2  —  (m  +  n)^ 

25.  (a  — m)(a  — w)  —(a  —  m)(a  — p)  —  (m  —  a)  (n  — i?). 

26.  (m  -{-  n  4- p)^  —  m  (n  +  j9  —  m)  —  n  {m  +  p  —  n) . 

27.  (3m4-l)(3m-l)  +  [mn  -  Jl-m  (2w-9m)J]. 

28.  3  a  -  [2  a  +  3  (a  -  1)  -  2  (3  -  2  a)]  -  4. 

29.  2[3a-4(2a-l) -3(-2a-^r:^)]. 

30.  116-5[3-2  J8  +  3(4-2[8-3T^])|]. 


26  SIMPLE  EQUATIONS 

SIMPLE   EQUATIONS 
Exercise  12 

Solve  the  following  equations : 

1.  3x  +  4:  =  2x  +  5.  6.  2(x-\-S)  =  A  +  (x-2). 

2.  5aj  +  9  =  2a;+.15.  7.  5  (x-2)  =  3  (a;+l)-l;i 
Z.   Sx-4:  =  x  +  12.  8.  7aj-(;^j-3)-12  =  2a-. 

4.  4a.' +  3  =  a; +  6.  9.    3  (x  +  2) +x^  =  5 -\- x\ 

5.  5a; +  7  =  2  0^4-9.  10.    (a^+l)  (a:+2)  =a;(a;+l). 

11.  4  +  5  (a;  +  2)  -  9  a;  =  (a;  4-  2)2  -  x\ 

12.  (x  +  2)(x-5)  =  (x  +  4)  (x  -  1). 

13.  (a;-l)(a;H-3)-2(a;4-l)(aJ-5)+a;2^0^ 

14.  2(a;2  +  2a;  +  l)  -  (a;  +  2)  =2a;2  +  6. 

15.  (x  +  4:y  +  (x  +  iy=  (a;  +  3)2+2a;(a;+l)-a^. 

16.  3  (a;  4-  5)  (a;  +  2)  -  («  +  3)  (a;  -  1)  =  2  ar^  -  (a;  +  7). 

17.  (4a;-l)(a;  +  3)-4a;2_(_i0^_^3)_^(3^Q        ^ 

18.  2[a;  +  a^(ic-3)H-l]=(2a;  +  5)(a;-l). 

19.  5S2(a;  +  l)  -(a;  +  3)S  =3[a;  +  2Sa;-5(3-a;)S]. 

20.  2  [3  a^  4-  (aJ  -  2)  (a;  -  1)]  =  3  [2  a^  +  (a;  -  3)]  +  2  a^. 


21.  3[5a;-(a;4-3  4-2a;-l)]  =  3a;-4  53a;  +  2(a;-l4-3a;)J. 

22.  [(a;-2)(a;4-l)  +  (a;  +  3)(x4-2)]  =  [(a^4-3)(aj-5) 

^(x-5)(x  +  l)l 

23 .  (a;  4-  2)  (2  aj  4- 1)  (3  a;  4-  3)  =  (6  a;  -  3)  (a;  4- 1)  (a?  +  3). 

24.  aa;  +  a  =  4a.  27.    2cx  +  d  =  4:G^-\-d. 

25.  2ax-{-c  =  5c.  28.    ax  —  (a-\-b)  =  3a+b. 

26.  4aa;  — 5c  =  5a  — 5c.  29.    2(a  — a;)=8a. 


PROBLEMS  IN  SIMPLE  EQUATIONS  27 

30.  3(x-{-a)-\-2{x  —  a)  =  6(a  —  x). 

31.  (a  +  b)x-^(a  —  b)x  =  a^b, 

32.  5  a -\- (a -^  bx)  c  =  ac  —  bcx. 

33.  10(a  +  b) -\-3x=a-\-b  —  5x. 

34.  S(a  +  b)x  —  2(a  —  b)x  =  a-\-5b. 

35.  (x  —  m)(x  —  n)  =  {x  —  m  —  iif. 

PROBLEMS   IN   SIMPLE   EQUATIONS 

Exercise  13 

1.  What  number  is  that  which,  when  doubled,  equals  24  ? 

2.  What  is  the  number  that,  increased  by  12,  equals  27  ? 

3.  If  a  certain  number  is  increased  by  12,  twice  the  sum 
will  be  28.     What  is  the  number? 

4.  Four  times  a  certain  number  when  diminished  by  6  is 
equal  to  12  more  than  the  number.     What  is  the  number  ? 

5.  There  are  two  numbers  whose  sum  is  77,  and  the  greater 
is  13  more  than  the  smaller.     Find  them. 

6.  A  man  is  13  years  older  than  his  brother,  and  the  sum 
of  their  ages  is  49  years.     Find  the  age  of  each. 

7.  A  father  is  4  times  as  old  as  his  son,  and  the  sum  of 
their  ages  is  55  years.     Find  the  age  of  each. 

8.  The  sum  of  the  ages  of  three  brothers  is  85  years.  The 
oldest  is  twice  the  age  of  the  youngest  and  5  years  older  than 
the  second.     Find  the  age  of  each. 

9.  A  child  is  3  years  older  than  his  brother,  and  5  times 
his  age  is  3  years  more  than  6  times  his  brother's  age.  Find 
the  age  of  each. 


28  PROBLEMS  IN  SIMPLE  EQUATIONS 

10.  Five  years  ago  a  man  was  4  times  as  old  as  his  son,  but 
now  he  is  only  3  times  as  old.     Find  the  present  age  of  each. 

11.  A  man  bought  the  same  number  each  of  2-cent,  5-cent, 
and  6-cent  stamps,  paying  ^  0.91  for  the  lot.  How  many  of 
each  kind  did  he  buy  ? 

12.  Find  three  consecutive  numbers  whose  sum  is  39. 

13.  Find  three  consecutive  odd  numbers  whose  sum  is  33. 

14.  Find  two  consecutive  even  numbers,  the  difference  of 
whose  squares  is  52. 

15.  A  man  bought  a  number  of  horses  at  $  150  each,  twice 
as  many  cows  at  ^40  each,  and  3  times  as  many  sheep  at 
$  5  each.  The  lot  cost  $  1225.  How  many  of  each  kind  did 
he  buy  ? 

16.  How  can  you  pay  a  bill  of  $  80  so  as  to  use  the  same 
number  each  of  1-dollar,  5-dollar,  and  10-dollar  bills  ? 

17.  A  man  asked  a  farmer  how  many  cows  he  had,  and  was 
answered,  ''  If  you  gave  me  18  more,  I  should  then  have  twice 
as  many  as  I  now  have."     How  many  had  he  ? 

18.  A  man  sold  15  hens,  receiving  80  cents  each  for  a  part 
and  50  cents  each  for  the  remainder.  He  got  ^  9.60  for  all. 
How  many  were  sold  at  each  price  ? 

19.  Three  dollars  in  nickels  and  dimes  were  distributed 
among  42  boys,  and  each  boy  received  one  coin.  How  many 
boys  received  dimes  ? 

20.  Into  what  two  amounts  must  $  1700  be  divided  so  that 
the  income  of  one  part  at  5  per  cent  interest  shall  be  double 
the  income  of  the  other  part  at  6  per  cent  interest  ? 


REVIEW 
£zercise  14 

1.  If  a  =  3,  6  =  1,  c  =  0,  and  cl  =  l,  find  the  value  of 

a  —  (a  —  b)  +  \a  —  {b  +  c)l  —  [a  —  (b  —  c  —  d)]. 

2.  From  what  expression  must  you  subtract  the  sum  of 
5 a^ 4- 8,  3 a 4- 2 a^,  and  a^ -\-a—l,  to  produce  the  expression 
4a2-8a  +  3? 

3.  If  a  =  7n-\-n  —  2p,  b  =  m  —  2n-{-p,  and  c  =  ?i+p  — 2m, 
show  that  a  -j-  6  4-  c  =  0. 

4.  What  must  be  added  to  a^  +  a^  —  2  a  4-  3  that  the  sum 
may  be  —  a^  —  a^  4-  2  a  —  3. 

5.  To  what  expression  must  x*  —  3x^  +  2a^  —  x-\-5  be 
added  to  produce  a^  —  x  —  1? 

6.  What  is  the  numerical  value  of  the  remainder  when 
3a4-2c  — d  is  subtracted  from  4a4-3c  — 2d  if  a-\-c  =  d 
and  d  =  7? 

7.  If  A  =  a^-l-^4:0^,  B  =  -x-2x'-\-l,  and  C=2a^-i- 
2 0)2  +  a; +  1,  find  the  value  of  -  A-[B- (2  A- C)-\- C^ 

8.  Simplify 


4a- [- 6c- (- 54-26-3  (^) -4  a] -5- (4c -3  cQ. 
9.    Simplify 

i-[-i+!-i-(-i+?-i+(-i)s-i)n- 

10.    What  is  the  coefficient  of  x  in  the  reduced  form  of  the 
expression  (x  — 4a)  —  [2a  — Six  — 2 (x  —  a)j]? 

29 


30  BEVIEW 

11.  Multiply    m^  — 2m»  +  2m-  — 2m  +  l    by    m*  +  2m^  + 
3  m^  -{-2m-\- 1. 

12.  Multiply  a*b  -  a^b' -\- a^b^  -  ab^  by  a^b  -  d'b^ -\- ab^ 

13.  Multiply  ^a^  — I  tt2  + a  — 1  by  i a  — 1. 

14.  Multiply  0.1  aj2  +  . 04  0^4-0.5  by  0.1  x-^  +  4  ic  +  .05. 

15.  Divide   Gm^  +  ^/i^  — 29m2  +  27m  —  9    by    3m^  +  5wr  — 
7  m  +  3. 

16.  Divide  1.2  aj^  -  2.9  a^  + -9  a;^  +  a;  by  .3  a; -.5. 

17.  Multiply  by  inspection  (a -{-b  —  2)((i—b  —  2). 

18.  Multiply  by  inspection  (a -\- b  —  c)  (a  —  b  -{•  c). 

19.  Expand  by  inspection  (a  —  2b-\-Sc  —  d)\ 

20.  Divide  1  by  1  —  3  m  to  5  terms. 

21.  Divide  m'^  by  m  +  2  to  4  terms. 

22.  Simplify  (5  a  + 1)  (a  -  3)  -  (2  a  -  3/  _  (a  -  5)  (a  +  3). 

23.  Simplify  (2a-l){a-{-4:)-2a'-\3a-{-(2 a-l){a-6)l. 

24.  Simplify  5  a  +  (4  a  -  1)  +  «  +  3  («  +  1)  -  (a  +  3)  (a  +  1). 

25.  Simplify  4  a^  -  3  a  [a^  +  a^  -  (a  -  2)]  -  3 (a  +  1)  (a  -  7). 

26.  Find  the  value  of  (a  +  bf  —  (a  +  c)^—  2  (a  +  6  +  c)  when 
a  =  l,  b  =  2,  and  c  =  0. 

27.  Find  value  of  -^  + V6^-4ac^  ^j^^^  ^^5^  5^_11^ 
and  c  —  Z. 

28.  Find  value  of  -?>-V&^-4ac^  ^^^^^^  ^^^^  &  =  -ll, 
and  c  =  — 3. 

29.  [(a;  +  a)2  4-  5(a;  +  a)  +  4]  --  [(a?  +  «)  +  !]  =  ? 

30.  [5(a;  +  m)4-3][5(a;4-m)-3]  =  ? 

31.  Solve  2a  — 3a;(a  +  c)  =  5a  +  3c. 


FACTORING  31 

32.  Solve  x'-(x-\-  af  =  (a  +  If. 

33.  Solve  a'-\-{x-l){x-2)  =  x'  +  (a-l){a  +  l). 

34.  Solve  4(a;  +  4)(a;-3)-2(a^-2)  =  3(a;  +  l)(x-4)-cc2. 

FACTORING 

Ezercise  15 
Factor : 

1.  a^^a^  +  a.  10.  Ga-^-Qa^  +  Sa. 

2.  m^  — m^H-m.  11.  5  m  —  10  7)i^  + 15  m\ 

3.  2m  +  4m2  +  6m».  12.  12  m' -  18  m^  +  24  m«. 

4.  5c2+10cH-15.  13.  5ac— 10  6c +  5  cd. 

5.  a24-4a^  +  6a*.  14.  4  a^c  _  10  ftc^  +  6  ac. 

6.  3a^  +  9aj^-6a^.  15.  6  a^y -\- S  xy^  -  9  x^. 

7.  10  a.-3  -  12  ar^  +  13  £c.  16.  m^ -S  m'  +  4.m^-m^ 

8.  8a:^-12x»-16ar2.  17.  a^c  -  aV  +  aV  +  ac^ 

9.  4a3-f  8a2  +  12a.  18.  S  a'-\-^a^-2  a^ -{-6  a^ 

19.  a'b-a%^-{-a^b^-ab\ 

20.  4a*6-12a^62-16a^63  +  8a*6^ 

21 .  15  ay  +  150  ay  -  225  a/  +  15  a/. 

22 .  48  7n^n^  - 144  mhi^  - 192  m«ri^  +  240  m'^'n^K 

Exercise  16 

Factor : 

1.  a2  +  4a  +  4.  4.    a^-20a;  +  100. 

2.  a2  +  6a  +  9.  5.    a2-18a  +  81. 

3.  a2-8a4-16.  6.    4m2H-4m  +  l. 


32  FACTORING 

7.  0,2  _^  22  a +  121.  18.   25  a' -  SO  a -}- 9. 

8.  16c2  +  8c  +  l.  19.    9a2-30a  +  25. 

9.  a2-36a  +  324.  20.   49c2-84c  +  36. 

10.  36a2-12a  +  l.  21.    16  a^b^  +  S  abc -^  c". 

11.  9-6a  +  a2.  22.    a^  +  lSa^  +  Sla. 

12.  m2  +  42m  +  441.  23.    36  a^  +  60  a'c -\- 25  a^c^. 

13.  a-^>--14a6  +  49.  24.    144  m^  -  240  m/i  + 100  w^. 

14.  ify  +  32a'2/  +  256.  25.    121  x^  -  374  a.-^  +  289. 

15.  cv'cH^- 10  acd-\- 25.  26.    625  m^  -  50  ?7i  +  1. 

16.  64-16m?z  +  mV.  27.    (a  +  6)2  +  2(a  + 5) +1. 

17.  4a2+12aH-9.  28.    (a  +  by -^6(a  +  b) +  9. 

29.  (a-c)2-6(a-c)H-9. 

30.  25(a-m)2-70(a-m)+49. 

Exercise  17 

Factor : 

1.  a'-b\                   11.  64a;2-25.             21.    81a^-49a. 

2.  a2-4.                    12.  a*-l.                   22.    81mV-16. 

3.  a2-4m2.               13.  a^-Slc^.              23.   324- 256  a^/. 

4.  c2-9fZ2.                14.  aV-25.              24.   289  - 16  m^. 

5.  0^2-16.                  15.  9a*-4a«.            25.    (a +  6)^-1. 

6.  m2-49.                 16.  6a^-24a.           26.    (m-ri)2-4. 

7.  9  0^-16.               17.  3  ar^- 75  X.            27.    7ri'-(n-^py. 

8.  25  a^b^ -9.            18.  a^6^-81.               28.    9a''-(b-cy. 

9.  36c2-25.             19.  121  a^- 49.           29.    4(a  +  &)2-c2. 
10.    36c2d2_9^            20.  64a*-a8.             30.    16(x-yy-9. 


FACTORING  33 

31.  4a2_9(a  +  l)2.  34.    36  (a  +  &)' -  49  (m  -  w)^. 

32.  9  a^- 16  (a +  2)2.  35.    {a?  -  Wf  -  4.  {o? -{- Wf 

33.  2^{a-hf-{c-\-df.  36.    100  -  (a -f- 6  +  c)^. 

37.  81(a  +  6)2-4(a  +  6  +  l)2. 

38.  9(a2  +  &  +  c)2-16(a2_&_c)2. 

Exercise  18 

Factor : 

1.  a^  +  2ah-\-h'^-c\  11.  (? -^  cd-\-^dr -lQ>m\ 

2.  m2-2mn  +  n2-p2^  12.  4«6-4  a^-f- 1  -  61 

3.  x'^  —  h'^-[-y'^  —  2xy.  13.  in^  —  A  mn  -  9  7n-n^  +  4:  n\ 

4.  m2  +  ?i2-j92^2mn.  14.  9  a^-2522  +  16/  + 24  a-?/. 

5.  m2+n2-^2_2y^n.  15.  4a2  +  12a6-9c2  +  9&l 

6.  2m  +  n2-m2-l.  16.  20 mn -^ p^ - 4: m^ -  25 n\ 

7.  l_a^-2a;y-2/^.  17.  4a2  +  a*- 4a3-l. 

8.  ar^_4a2-4a-l.  18.  5  ar'-S-Sa^-lOa^. 

9.  2mn  +  mV  +  l— p2.  19.  8ac  — 4  a2_4c2  +  4. 
10.  ci--c2-l-2c.  20.  m^  +  16n^-16-\-Smn. 

21.  a--^2ab  +  b-  —  7nr  —  2  mn  —  n\ 

22.  4a2-4a  +  l-9a.'2-|-6a;2/-/. 

23.  9a'-30a  +  25-4.b'-4:b-l. 

24.  a2-c2  +  62_^2_2a&_2cd 

25.  a%^  ^  10  a'mhj  -  n^  - 1  _  2  n  +  25  /. 

26.  711^  —  n'^  —  x^-\-if  —  2  (my  —  wic). 

27.  25  a^  +  1  -  1 6  a^  _  9  c^  -  10  a2  _  24  a^c. 

R.  &  8.  EX.  IN  ALG,  —  3 


34 


FACTORING 


28.  5a2  +  562_5m2-10(a6  +  m7i)-5w«. 

29.  -12a6  +  2  +  24a25_i862_^18a262_8a*. 

30.  3a2  +  1262_l2a4_i47  52_34^2^_-|_2«5. 


Factor : 

1.  m^  +  m^n^  +  n*. 

2.  x^-lx^y'^-\-y\ 

3.  iB*-5a^3/2^4^^ 

4.  m*-23m2  +  l. 

5.  a^-79a2  +  l. 

6.  m*-171m2  +  l. 


Exercise  19 


7.  25a^  +  66a262  +  496^ 

8.  49  X*- 11  a;22^2_^  25  2/^ 

9.  16a;*-73a^  +  36. 

10.  49a^-74a262^25  6*. 

11.  289m^-42mV  +  169w^ 

12.  16a*-145a262  +  9  6*. 


Factor : 

1.  a2  +  3a  +  2. 

2.  a2-a-12. 

3.  a2-9a  +  20. 

4.  ar'  +  5a;-24. 

5.  a2  +  18a  +  17. 

6.  c2-llc  +  24. 

7.  m2-19m  +  88. 

8.  c2-9c-22. 

9.  a;2^5a._14. 

10.  a2-3a-28. 

11.  iB2^9a.^i4^ 


Exercise  20 


12.  2/'-ll2/  +  28. 

13.  ar^-9a;  +  14/ 

14.  c2  +  42c--43. 

15.  m2-4m-165. 

16.  2/' +  12  2/ -108. 

17.  aV-21aa;-46. 

18.  a262  +  13a6+40. 

19.  aW  + 21  am -130. 

20.  c2d2  +  9cd-52. 

21.  mV-2??i2n-35. 

22.  ic2«2_20fl;;2-69. 


FACTORING  36 

23.  x^f-xy-12.  34.  m*  — m*-156m^ 

24.  a;^-13ic2  +  36.  36.  a^+ (a  +  &)a;  +  a6. 

25.  ic^  —  9  a^  —  22 ic.  36.  a;^  +  (m  +  n) a;  +  mw. 

26.  a;*  — Sar'  — 9.  37.  a^ -\- {c  +  d) a -{- cd. 

27.  a^  — 7a^  — 78  a.  38.  x^  —  {m-\-n)x-{-mn. 

28.  a262_6a6-187.  39.  a;^^  (a4- 26)a;  +  2a6. 

29.  16-6a-a2.  40.  a^+ (3a  +  2  6)a;+.6a6. 

30.  18-19c+c2.  41.  iB2_(a-6)a;-a6. 

31.  147  - 46 a--^ -  a;«.  42.  x^-ax-hx  +  ah. 

32.  90m24-13m^-m«.  43.  a^- (3m-2)a- 6m. 

33.  a:3_^i0a:2_963.^  44^  a^-\-{m-2m7?)ay-2m?ii?y\ 

Ibcercise  21 

Factor : 

1.  2a2  +  5a  +  3.  14.  8a2-30a-8. 

2.  6ar'-a;-2.  15.  24m2-14m-49. 

3.  2ar^-3a;-9.  16.  2 a;^ -f  7 a; - 15. 

4.  2a2  +  7«  +  3.  17.  18a2-f-9a-2. 

5.  8a^+^a;-3.  18.  40 ar^ - 61  a;  +  7. 

6.  6m2+^m-5.  19.  8m2  +  2m-3. 

7.  15ar^  +  lla;  +  2.  20.  ^b  a"  -  IS  a  - 12, 

8.  7ar^-41a;-6.  21.  6a2  +  25a-9. 

9.  6a2-29a  +  28.  22.  8m2  +  5m-3. 

10.  3a2-19a  +  6.  23.  4.2 x" - 11  x - 20. 

11.  12c2  +  17c-5.  24.  16m2-67n-27. 

12.  6/-y-12.  25.  12y^-y-20. 

13.  3m2-llm  +  6.  26.  2x'-4.x-12^, 


36  FACTORING 

27.  12  m^  -  7  m^n- 12  m?il  36.  20  a^^^  -  9  a^ft^  -  20  a&. 

28.  6m3  +  29m2-22m.  37.  16  0^^2  +  2  ccZ- 3. 

29.  8x^-26^^2  +  18.  38.  75  a  -  210  aa;  +  147  ax^ 

30.  aW-9aW  +  20a2m.  39.  48  a;^  - 176  x*  +  65  aj^. 

31.  26a34-197a2  +  l5a.  40.  55x-«-x^-2x3 

32.  52a2-153a-52.  41.  2(a  +  l)2  +  3(a4-l) +  1. 

33.  3m2-30m  +  63.  42.  3(a  + 1)^- 8(a  + 1)  +  4. 

34.  12x^-25x2  +  12.  43.  2(a  +  l)2  +  5(a  + 1)  +  2. 

35.  x«-x*-42x3.  44.  2(a-2)2-5(a-2)-3. 

45.  3 (m  -  1)2  -  11  (m  - 1)  ri  +  6  nl 

46.  10  (a  +  6)2  -  11  (a2_  62) +  3  (a -6)2. 

Exercise  22 
Factor : 

1.  m^-n\  8.  a365-8.  15.  343 «%« -  729  7i«. 

2.  m3-27.  9.  27- 8  c^c^^^  16.  {m^nf  +  3?. 

3.  8a3  +  l.  10.  7MV-343.  17.  (c  +  d)»-8. 

4.  27-8c3.  11.  64a3  +  125.  18.  (a +  1)^  +  64. 

5.  27a3  +  8.  12.  8a«-27.  19.  27-8(a  +  2)3. 

6.  64- 125  c^.  13.  64a»  +  c^  20.  64  (a  +  6)^  -  27  a^ 

7.  125a3+2763.  14.  125mi2_^i5^  gl.  (a  +  1)^ - 8 (a  +  2)^. 


22.  a^-h\  26.  a}''-2^.  30.  (a -6)^-1. 

23.  a^-16.  27.  4m^-81.  31.  (a-6)«-8. 

24.  81 -m^  28.  a^^'-w}''.  32.  (a +  6)^ -256. 

25.  a«-64.  29.  m^^-'n}^  33.  (2a+l)^-16(2a-l)^ 


FACTORING  B7 

34.  m^  +  8.  38.  x^'^-^y^^  42.  x^^  +  1. 

35.  m^  +  1.  39.  x^  —  y^  43.  m«  +  27. 

36.  a^  —  l.  40.  a;^  +  2/^  44.  64a«  +  l. 

37.  3^ -{-32,  41.  i»8-^.  45.  a}^  +  7n^. 

Exercise  23 
Factor : 

1.  m^  +  mn  +  mp -{- n2).  13.  m^  + 5m2  +  2m  +  10. 

2.  ab-\-a-\-7b-\-T.  14.  m^w  —p^mn  -f-  mic  —  p^a;. 

3.  2c7n-3dm-\-2c-Sd.         15.  a^- 1 +  2(a2- 1). 

4.  ay-ab-bx-\-xy,  16.  a^  (a^  -  9)  -  a  (a  +  S)^. 

5.  a:3_^^_^a.^l^  17.  5(a^  +  8)-15(ic  +  2). 

6.  xy-2y-x^-{-2x.  18.  a;^4-2a^-8a;- 16. 

7.  a^-a^  +  a-l.  19.  2a2(a +  3) -Sa^-Sa  +  S. 

8.  m^-n^-m-n.  20.  c^-\-4:C^-S. 

9.  a^-a^-a  +  l.  21.  m^-lOm-SO. 

10.  6a3  +  4:a2_9a-6.  22.   4a3-39a  +  45. 

11.  m^  +  m^-m-l.  23.    a3  +  9a2  +  ll  a_21. 

12.  15 ax-2llay+9bx-12  by.   24.   a'^  +  3a2+ 3a  +  2. 

25 .  am  -{- an -\- ap -{-  bm  -\-bn-]-  bp. 

26.  m^ {n  —  x)  -{-  m{n  —  X)  —2 (n  —  x), 

27.  m^  +  ?/^  +  m  — 2m2/  — 2/-^6. 

28.  (a-l)(a  +  l)  +  (a;-l)(a-l). 

29.  (a  +  l)(aj  +  2)-(a  +  l)(y  +  2). 

30.  a^-l-xia-l). 

31.  m^  +  l  — ^>(m  +  l). 

32.  (m  +  l)(m2-4)-(m  +  l)(m  +  2)-m-2. 


REVIEW 

Exercise  24 
Factor : 

1.  6a2-fl9a  +  10.  18.  24:  a^b^  -  36  b*  -  SO  ab\ 

2.  2x'-6xy-U0y\  19.  7a;^-7a;. 

3.  a*-\-a.  20.  m^~m'^-2m^ 

4.  c*-^(^d^  +  d',  21.  m*  +  n*-23mV. 

5.  m^  —  m^  —  SOm.  22.  42/  +  a;2  — 1  —  4^2 

6.  c*  +  cc?8.  23.  (2m-5n)2-(m-27i)2. 

7.  a^  +  2xy-ix^-y\  24.  4a%2_4ci2^4^8^3^3_3^^6^ 

8.  a;4_iB3a  +  iK2;3_^2^^  25.  25a^b^c'-9, 

9.  a^-oc^y  +  xy^-y\  26.  a^-Sa-a^  +  S. 

10.  1  —  m^  —  2  mTi  —  n^.  27.  a;^  —  /. 

11.  c^  +  (^4_-|^8g2^2^  28.  m^-rn?n-\-mn^-n\ 

12.  m2wy_^2p_^?p_^j  29.  81m«-16m7i^ 

13.  6(?-c{d-V)-{d-Vf,     30.  72aH5a2_i2a. 

14.  a2-c2-4c-4.  31.  l  +  aa;-(c2  4-ac)aj2. 

15.  d2  +  3cf-d^-3ci.  32.  54-16mW 

16.  8a^4-27a;2^  33.  4.^f-(^^y^-zy. 

17.  JK«  +  2/3,  34.  49a^  +  34a^2^H-25y. 

35.  (a;  —  m)  (2/ —  n)  —  (a;  —  n)  (y  —  m). 

36.  15m'-14m^-8m3. 

38 


REVIEW  39 

37.  (m-n)(2a2-2a6)  +  (n-m)(2a6-262). 

38.  (a-l)(a-2)(a-3)-(a-l)  +  (a-l)(ci-2). 

39.  (2c2  +  3d2)a  +  (2a2  +  3c2)d. 

40.  d^  +  d^-d-^l,  62.    2m3n-16ni^ 

41.  x^-64:.  63.   64a;^-a;. 

42.  1000  + 27 c«.  64.    (a^+a -l)2-(a3-a-l)l 

43.  5a3-20a2-300a.  65.   a;^-27ar^  +  l. 

44.  mn— pr+j97i  — mr.  66.   Slx'^—y^z^^. 

45.  c2-2cd  +  d2-l.  67.   5cd-12d'  +  2c'. 

46.  4-9(a;-32/)'.  68.   72 ic^ _|. gg ^ _ 40 ^.^ 

47.  a'i»^-a^-a^-2ax.  69.   3iB*4-192a;/. 

48.  ISf-^Sy^-lSy.  70.   3c3-12c3d2_4^2_^l^ 

49.  m*  — 2mw«-n*  +  2m»w.  71.  pY-277^. 

50.  a;^  +  125a;/.  72.    mV-a^^-m^  +  l. 

51.  m3  +  m2w  +  2mn2  +  2?i3.  73.   aj^  -  25  (a;  -  3)2. 

52.  24a^-5a;-36.  74.   a^-c^ 

53.  72a^  +  ^x  —  4:5.  75.   5m^  —  5mhi~5mn  —  5m. 

54.  (a2-6y-(a2-a6)2.  76.    m^-27m2  +  162. 

55.  ac2  +  7ac-30a.  77.   9  a;^  +  68 a.-^ -  32. 

56.  x^y^  —  a^  —  2/^  + 1.  78.    1  —  m^n^  —  ph-^  +  2  mnpr. 

57.  a;*  +  4a^-8a;-32.  79.   m^-5m2  +  4. 

58.  5cV  +  35ca^-90a^.  80.   m^  -  m^x  -  m -\- x. 

59.  ac  +  cd-ab-bd.  81.   24 a.-^  +  43 a;^ - 56 a;. 

60.  a^  —  m^ -f- a;  —  m.  82.   m*—(m  —  6y. 

61.  9  a;^- 66*2  +  25.  83.    Sa}^  +  a7n}'. 


40  //.  C.  F,   AND  L.  C.  M. 

84.  24:€rd'-A7cd-75.  87.    a^c  +  3ac'-3a^-(^. 

85.  16a^-/-9  +  6y.  88.    c^  -  64  c^  +  64  a^  -  a^. 

86.  12a3  +  69a2  +  45a.  89.    a^  _  a^ft^  _  52  _  ^^ 

90.  2a;3^3^_3^^3 

91.  10  (??i  +  c)2  +  7  a(m  +  c)- 6  a^. 

92.  100  +  10a;^-25x«-ic2. 

93.  (m+py-l-2(m+p-\-l). 

94.  a2  +  2a-c2  +  4c-3. 

95.  (2m-3)2-6(2m-3)?i-7w2. 
'    96.  x'6-13a^4-12. 

97.  m2  +  7i2_(l  +  2mn). 

98.  (c-2d)^-9-3(c-2d  +  3). 

99.  6c3_25c2  +  8c-16. 

100.   x^y  +  y^z-{-  xz^  —  x^z  —  xif-  —  2/2^. 

HIGHEST   COMxMON   FACTOR   AND   LOWEST 
COMMON   MULTIPLE 

Rzercise  25 
By  factoring  find  the  H.  C.  F.  and  L.  C.  M.  of: 

1.  a^-fe^  a^-h\  o>-h\ 

2.  a^-ab',  2a*-2a'b%  a^-2a^b  +  ab\ 

3.  a^-16,  a'-a-2,  a2-4a  +  4. 

4.  m2-3m  +  2,  m2-m-2,  m2  +  m-6. 

5.  «2  +  a;-12,  cc2-4a;  +  3,  ar^  +  2a;-15. 


H.  C.  F,   AND  L,  C.  M.  41 

6.  o?-^2o?-l^a,  a?-6a?-\-Qa,  o?-2a^-Sa. 

7.  x^'-^x'  +  l,  2it?  +  2x'-2x,  a^H-2a^-l. 

8.  a;4  +  a^_6,  3a^H-6a^-24,  aj^-lOx^  +  lG. 

9.  2x'-nx-^0,  3a^-25a;  +  8,  x'-x-m. 

10.  5m3-5n^  15(m-7i)^  10  m^  -  20  m?i  +  10  nl 

11.  m*  — n*,  m^  +  m?n  —  mn^  —  n^j  m*  — 2mV-f  n*. 

12.  5a3  +  406^  7  a«  +  28  a^d  +  28  aft^  ^  a' -12  a?h\ 

13.  c^-d^  c^  +  d^  c^  +  d^^  (?  +  2cd-\-d\ 

14.  12a^-30a;-18,  27 a^ - 90 a;  +  27,  15 ar' - 42 a; - 9. 

15.  mn  —  mp  +  2n  —  2p,  m^H-6m^4-12m  +  8. 

16.  a;'' H- a^2/^  4- 2/^,  3?z -\- x^v -\- xyz -{- ocyv -{■  yh -\- yH. 

17.  12(m«-n«),  18(m*-n^),  24  (m^  -  mn  +  n'^). 

18.  a^-h^-.o?h^ab,  a? -\-W -a?h -al)",  a"  -  ab^  +  b*  -  a^b. 

19.  a^-3a'-^a-\-12,  a'-lSa'  +  Se,  a^  +  ^a^- 9a -^18. 

20.  a^  +  3a^-\-3a  +  2,  a'-Sa-S,  a^ -{-Sa^  +  a-2. 

21.  a3-22a  +  15,  a3  +  6a2_25,  a^  +  13a2  +  36a-20. 

f 

Exercise  26 
Find  the  H.  C.  F.  and  the  L.  C.  M.  of; 

1.  a:3  +  4a;2  +  7a;  +  6,  a^  +  4ar^  +  a;-6. 

2.  a:^  +  6x^-\-llx  +  12,  x^ ^2a^ -6x-\-S. 

3.  2a;*-a^  +  a:^-a;-l,  2x^  +  Sa^ -x^-^x  +  1. 

4.  3a;^4-3a^-3a;-3,  4a;*-4a^-8a^  +  4a;  +  4. 

5.  6ar'  +  19ar^  +  19a;  +  6,  4. x^ -{- S a^ -\- 5 x -\- 3. 

6.  a;^  +  3«3  +  5a^4-4a;  +  2,  a;^  +  3a.-34- 6a^  +  5a;  +  3. 

7.  2m*-{-m^-9m'-\-S7n-2,  2m^- Tm^H- 11  m^- 8m  +  2. 


42  FRACTIONS 

8.  2a*  +  5a3  +  2a2_a-2,  6a«  +  3a^  +  6a»-3. 

9.  3a3  +  14a2-5a-56,  6 o? +  10  a" +  11  a -\-^^. 

10.  4m*  +  3m3-6m2-29m  +  30,  A  m*  -  7n^  -  IS  m^ 

+ 14  m  -  5. 

11.  4a«  +  14a^4-20a3  +  70a2,  6  a«  +  21  a^-12a4-42a3. 

12.  a^-^a^b-\-6a%^-ah^-6b\  a^ -S  a^b-\-3  al/ -2b\ 

13.  c^-2c3-7c2-|-8c-10,  c^  +  c^-9c2  +  10c-8. 

14.  2  m''  —  m^n  — 11  mV  + 17  mw^  —  7  n'^,  m*  —  2  m^/i  —  m^n^ 

+  4  mn^  —  2n\ 

15.  a;'^4-2a;--10ic-21,  «34.4a^_2a;-15,   a^4-2cc2_7  3._-{^2. 

16.  2aXa'-2a^-7a'-\-16a  +  7),   5 a\a^- 5 a'- 23 a -S), 
6a(a3-6a2-26a-9). 

FRACTIONS 

I.   TRANSFORMATIONS 

Rsercise  27 

Reduce  to  lowest  terms  : 
V     a^  +  27  ^       r2a^-x-6         ^  a;^-81 


0^2-9                          12i»2-13a;  +  3  aj^+lSic^+Sl 

Ax'  +  ^x  +  l                3?^ -6  16  -  (m  +  ny 

5a;^4-5      '          '    6m^-24*  *     {m-Af-n' 

2a^-  2a -12  a44.ft3^^2_^q_|_]^ 

•    a'j^2d'-a-2'                              '  a'-l 

^     y?  +  m.r  4-  »-«  +  'nin  --     7j^^— 64 ^ 

a;2-h3mx  +  2m2  *                        *  m^+2m3-8m-16* 

4m^— 8mnH-4n^                12  m^ — m^^^  4-  m^n — ti^ 


FRACTIONS  43 


14. 


15. 


16. 


6  ac  —  2  ad  —  3  ?)c  +  hd 
9  ac  —  3  ad  H-  3  5c  -  hd 

(m  4-  ^)^  +  7  m  +  7  n  + 10 
w?-\-2mn  +  'n?-  4(m  +  n)  — 12* 


,^     10a^  +  5a;^-105a:3  ^,     a^4_7  ar^4.12  a;  +  4 


24  a;^ 

-64a;3- 

-24iB2 

m'- 

in^  —  ^m 

+  3 

2' 

m^  —  m  — 

1 

:^- 

lla;  +  6 

18. """^    »  22. 


8m-3 


19.    ^^ '  Jl'  23. 


a^_l_2aj2_7a;_2 


x3_20a>+33  a;*-14ar^  +  l 

2-5a-4a^+3a^  2^  5m^-5mV 

4-|.4^+9a2+4a«-5a**  '   2m«-2m2w  +  2mn2* 


Exercise  28 

Change  tci  mixed  expressions  : 

x-\-2       '  '  x^-^x-S 

B2_12a;-47  ^     4  a^+12  a;3_^^4 


2. 


a;  +  3  2a;3  +  7 

a^_2a^4-4a;-l  ^     fl^-a^  +  3a;  +  2 


iC- 

-3 

3a3 
a  —  b 

9. 

m^-\-2 

10. 

m^  +  m  —  1 

x'-\-3 

x'-^x-l  '   x'-{-2x  +  2 

12 

2  +  a;-a^ 


7.   .^.  9.      „     ^     ..  11.  "^ 


19  1 

12.   — =—  to  4  terms. 


44  FRACTIONS 

Exercise  29 
Change  to  improper  fractions : 

,  ^      x  —  1  e        2        2  I  in^nHm  +  n) 

2.    a  —  Sc-\ — •  6.    (c-\-d)^—^ ^* 

a  +  c  c-\-d 

4.    1 


a^  +  ft^ 

••    ^       ^  '  ^      a  +  1 

(a-^by 

8.      ^^(^'  +  3)                2. 

9.    m^  — 

2 

mn 

+2 

m 

-  71^)  4-  m  V 

2m?2+7i^— »^       ,„     -,    ,   /  9  I       I       1    \ 

10.    m-{-n—p — ^-      13.    l  +  (ar  +  a;H • 

m  +  wH-p  \  x^lj 

!b2+3x-2  L      V   ^    /- 

12.   a+2-^+6^+a'.  15.   (''-^-A+^  +  2. 

a  +  4  V="  +  l        / 

ie.l-[.-{2..^-^i)i--.)]. 

17.  3m-r("^  +  2X'^-^)-(-»^  +  ^)1-(«t  +  2). 

m  +  3  ^  ^ 

18.  a'+r-'»6-a'+(-a'  +  «6)-&n  I  „      ^, 

a  —  6 

19.  (a-6)»-r(a  +  6)  +  3«H&-a)+a(«'-a)-n 


Collect : 


FRACTIONS  45 

11.  ADDITION  AND  SUBTRACTION 
Exercise  30 


1.    5^4-3^  +  ?.  12.    ?  +  -^+      ^ 


3        4       2                                  a     a+1      a-1 
2     4a;4-l  .  2a;  — 1  ^^^    __a ^  _^      1 


4  3  a-1     a  +  1      a'-l 

2a;  +  l      a?-l  14     3  5  2  m- 3 

'      3a;2         2a;3  '  *   m     m-1       m^-l* 

,1.1  ,,^         2  3.1 


a;  (a;  4-1)      a;  (a;  —  1)  3  —  3  a-     5  —  5a     l  +  a 

2      ,      1  1  ^^m  —  n.n—p.p  —  m 

5.     -H -— r -•  lb. 1 i 

a;  —  1      a;  +  l      ar  —  1  mn  np  mp 

g  +  l  '  g- 1  --,     4a  — 6        18  a6     .  4a+ft 
*   a-1      a  +  l'  '   a+2  6     a2-4  62     a-2  6' 

7.    a  +  2     a-2  18.    "^  "^  ^         ^ 


a-2     a-\-2  a^-1     a;-l 

„     5a  +  l      2a  +  l  ,«     c^  +  c?^  c^  d^ 

o.    — -— •  ly. 


aH-3        a-3  cd         cd  +  (P     cd  +  c^ 

9.    ,^±4_  .  ^^.  20.    -^-^^+-^  +  .    1 


(a-c)2     a'-c'  a^^x-\-l     x-1     sc^-l 

6 


^Q     (m  +  ny_(m-n)\  ^^     1 1__^ 

m  —  n  m-\-n  a^+2x-{-4:     x—2     oc^—S 

11.    -A_  +  ^_  +  ^.       22.   2^2m  _^   m+l  1 


m  +  1     m  — 1      m  — 2  (m— 1)^     (m— 1)^     m—1 


46  FRACTIONS 

23  a+r.  a-m        ^^_      a^^_a_^ 

ar-{-am-\-7rr    ar—am+mr  a -^2  a  — 2 

25.    -^  +  2--i-.  28.    3c--^  +  -i-. 

x+1  x—1  c+1      c— 1 

30.   (ra+-J^\-U+-J^\   32.    2-^+^+^ 
\^        m—nj     \       m+nj  \^—y    ^+y)     x^—y^ 

33^     K  +  ny  _/m_^7i_^^ 
mn{m  —  nf     \n     m 


31.   ^±1_24-^^- 
a;4-2  a;-4 


34  a?  — 3a         ,       9aa;       ,        1 


ic2_3aa;  +  9a2     a^  +  27a3     a;4-3a 
35.    4J^  +  ^  +  J^.  36.    ^+      3  1 


a^-1     1-a     a  +  1  2-a     a-\-2     a^-4: 

3^     g  +  l  4         g-l 

a  +  3      9-a2     3-tt* 

38.   1  +  -J^+     2  2  3 


X     a;  — 1      l—a^     x-\-l     x-\-7? 


39.    _A___1__^+       -8 


3m4-15     125-5m=^     7m-35 
40    ^  —  5  I  ^  +  5  .     21m 


5  +  ^1     5  —  m     m^  —  25 
.,     1  ,       20a       ,       2  1 


a     l-lGa''     8a-2     a  +  4a2 


42. 


1     +6-^  + 


FRACTIONS 
1 


47 


a^3  '         S-a  '  9-a2 
2  Sx'  4 


43.    3x  + 


44. 


45. 


46. 


47. 


48. 


49. 


50. 


51. 


52. 


53. 


54. 


x-S     x-\-S     9-«2 
m  +  1  1         m^  H-  m  —  2 


m^  +  m  +  l      m  —  1         1  —  m* 

c  — a?     c  +  <?  _  •     (d  —  c)^ 
c  +  c?     c  —  a;     {c  —  x)(c  —  d) 


x  —  Sa 


3ax~2y?  1 


ar^_3aa;  +  9ci2      a^  +  27a«      3a  +  a; 
a+1  ,  a+4 


(a-4)(2-c)      (a-l)(2-c) 


(m  —  n)  (m  — p)      (n  — p)  (n  —  m)      (p  —  m)(p  —  n) 


x" 


+ 


2/^ 


+ 


(a; -2/)  (a; -2;)      (y-z){y-x)      (z-x)(z^y) 

m  +  1 


m-1  2 


+ 


m^  —  m  +  1     m*  +  m^  +  l     m^  +  m  +  1 


-1— f-i L_l 

_^ [o 1 (     1        ,      ^    \\ 

x  +  b     \        x-n     \a?-25     x-5)] 

\2x  +  2-(^ l_^1_-2^_  (1_^__A_2^ 

-J— f-J— (^ ( 1 +^-M1 

1-x     [x-2      \&-x-i?     \2-Zx  +  ii?     x-X)]_ 


48  FRACTIONS 

III.  MULTIPLICATION  AND  DIVISION 

Exercise  31 
Simplify : 


1. 


m^  —  n^  m^ 


w?         (m  —  Tif 

^2-144      a  +  3 
a2_9       ^_^i2 

w?  —  n^    m^  —  mn-{-  n^ 

m-{-n 

m^  +  n^          m—n 

7nf-\-  mn  +  n^ 

(c  4-  df  ^    c'-cd  +  dj' 
c3  +  (Z3  *  c2-f-2cd  +  c«'* 


m2-2m  +  l 


(m  4-  2)  (m  -  1)     (m  +  3)  (m  -  2) 
'    m-\-n     \n      m) 


mn   /m  +  ^^  j 

7n-{-n\.     n  m 


— > 


c-d  '  c^-d^'  (?  +  df 
10.    (^^/-^j(^^2(^2/2_^i)  +  i)- 

a^-{-  {m  +p)a  +  mp     a^  —  n^ 
12.  fg-Sm  +  mAfl+i  +  i-T— — 4^— — l- 


FRACTIONS 


49 


13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 


3  (m  —  yi)       m^—  {p  —  5)m  —  5p  '  m-\-p 
m(m^-\-5m)     m^ -{- mp  —  mn  —  np     m—p 

a^  +  ac  +  ab  +  he  ^  g?  +  «/+  «cg  +  df 
(J?  -\- ac -^  ad -^  d/:    c?  -\-  od  -\-  ab  -^hd 


m-\-n  .m 


A  _^  fm-^n  _  m  —  rb\ 
i)  '  \m  —  n     m-\-nJ 

+  d     c-(A/6         3  Sy 

—  d     c  +  (ZyVc      c  —  d     c-\-d) 


m  —  n     m-\-n^ 

'c  +  d     c  — 
c 

m^-9   ^  (m-4)^  .  |^m^-7m  +  12     7/i  +  4^ 

*  m  +  3, 


6    a2-2a-3 


a^-g 


+  d     c  —  d\ 
—  d     c-^dj 


a^^a-2       a^-2a  +  l 
c2  +  d2   1^- 


4  m^  —  n^  —p^  —  2np_^  2m  —  n—p 
A  7n^  +  n^  —  p^ -^  4:  mn      2m-\-n-{-p 

a^-5a;-f  6  ^  a^4-a;-2  .  a;^-4a;  +  3 
a^^Sx  +  2'  x'  +  x-e  '  arH-4a;  +  3* 

(-•D(S-')H'-?> 

3  cd(d  -c)  +  (^-d^     r       dF\  .  c^(c-cg)^ 

(a_cy-&2     (g  +  &  +  c)'     (g  +  cf  -  b^ 
(5  +  c)2-a2*  la-i-by-c"'  a^-(b-^cy' 

Tin}  — 2  w^v?  -\-n^  m^  —  r^         ,  in 


m^  +  ri"  tti^-mH^^-n''  '  (m^  +  w«)  (m^  +  n^) 


4 


R.  &  S.  EX.  IK  ALG 


x{x  —  1)  + 1 


50 


FRACTIONS 


27.    f£_Ji-^-2Yl 2^Y 

\yz     xz     xy     xj\       x  +  y  +  zj 

28.  f™-i2iY^r('-+— -4Y-+— +4\ 

\n       mj      [\n       m        J\n       m        j 

29     (c  +  d)^-4     d'-{c-2)^  .  c^d-2 
'   4  -  (c  -  d)2     (^-{d-\-2f  c-d-2 

"■  [e-'-'-?)-e-'-^)]*[('-^)fe)} 

[_\xy     xz     yzj  xy         J 


Simplify : 
1 


1. 


—  X 


+  1 


lY.  COMPLEX  FRACTIONS 
Exercise  32 


a4-2 


4. 


a-3     "^ 


16      ^ 
a? 


8      10_3 
X*      a^      sc^ 


m  +  2 


2. 


3. 


m 

m-3  +  - 
m 

a  +  l 

4 

a; 

a^+l 

1 

1 

ajH-1 


x-\-l  — 


5. 


a;  +  4  — 


«  + 


a;  +  2  . 


X  — 


x-^1 


8. 


d'     c" 
i  +  i+1 

m     n     p 
n      p     771 


FRACTIONS  51 


x^  —  x  —  6 
10.    -o — 71 ?r-  11 


12. 


13. 


a^-x-6 

•     a.'2-6a;-7 

c—d     c+d 
c+c?     c  —  d 

c'-d'      ^ 

ic-df 
2           3 

a; -14 

a     &     a     6 


16.   1- 


4a 


^    2m      ,  ,      _J__\ 
Vl+J-  2m  +  V 


18. 


2m 

2 


1  — a; 


(f-')-(f-') 


I4.I     i_l  20.   a  + =— . 

a     h     a     h  a-\- 


1 4 .  •         1 

&  ,  a     h     a  a  +  - 


a 


ft^  +  a-2  3fa4.2)-2(a  +  l) 

a2_2a-3     . 

15-        2   ■    Q^_i_9~^'  21. 

tt2_4a4-3 


10     10  2^- 


a? 

a3  +  64 

a^- 

.4a«  +  16a2 
m  +  1 

m  +  1 

1 

5       5  m  +  l 

v2 


^^'       a?     ,  g^c  +  Z  *         ^  +  ^-^ 

^z-c+(^:r^2  100^+9^-63 


52 


FRACTIONS 


24.    2a  + 


3a  + 


4a-l 


26. 


1  + 


1  + 


l-x 


25. 


1  + 


4  wn 


(m-2nV 
4?nri 


1_^ 


27. 


1 

1 

1 

a-?> 

aH-6 

b' 

— 

a^ 

264-1 

a  —  b 


28. 


29. 


m^  —  71^        (711  -\-  7i)(m  —  p) 
m?  +  7i^    m^  —  mp  —  mn  4-  np 

\n''     m?i      my      \7r     mn     my 

l+3a     l-3a 
l-3a     l+3a 


3a     1 


3a 


l+3a     l-3a 


30. 


np 

mp 

1 

mn 

(m- 

-P)^- 

-ii^ 

1^_ 

np 

1 
mp 

1 

mn 

31. 


a/*^  +  mx  +  nx-\-  mn 
x^-\-nx  —  px  —  np 
a?  +  mx  —  nx  —  mn 
x^—nx  -\-px  —  7ip 


32. 


14- c3 


1- 


1-f 


8c3 


1-c 


1  + 


2c 


2c 


14-2C 


33. 


rx  +  yV  ^  /x-yy 
x-yV      /'x  +  yV 


34. 


m4-n  ■  m^  +  n^' 


m  —  n 


m' 


m^  4-  ^i^ 


+  n 


m' 


FRACTIONS 


53 


35. 


36. 


(m  —  n)  (m  +  p)      (m  -f-  n)  (m  —  p) 


{m  —  n)(m -\- p)      {m  +  n){m  —  p) 


37. 


3a;  — 1     X 

3      "^4 


^-l(x+^^-)-\-2x 


38. 


1  _  ^  +  ^  '  I  _  c^  +  (?" 


39. 


2mn 
4:mn 
m^  —  4  wm  +  n^ 


-r- 14  m?i  . 

\    mn  J 


40. 


i_i     i_jL,i 

'     ah     a"- 


a     b 


X 


+ 


11 


62     a6"^a2 


62 -a^ 


a'b^ 


41. 


r46(a  +  26)  J       (a3-86V      1 

|_     ft-26  JLa^-8q262_^l()6^J 


a2  4-  2  ty6  +  4  6^ 
a  +  26 


(m^  -\-  n^)(m  —  n) 


m^  —  mn  +  71^    m^  —  2  mr? 


(m  —  ny         m^  H-  mw  +  n' 


54 


FB  ACTIONS 


43. 


44. 


-  (»--)f'+-y 


^ 

1+ 


a 


45.  Find  the  value  of 1  when  a  =  x-\-l. 

a 

a^  —  b^ 

46.  Find  the  value  of  — when  a  =  ic  + 1  and  b  =  x  —  l. 

47.  Find  the  value  of  (m-iy-(m^-l)  ^hen  m  =  a  +  l. 

(1  —  my 

48.  Find  the  value  of 

a-\-m  ,  a  —  m       Sam     _.  „^  _      3a 

1 when  7n  =  — • 

a  —  m     a-\-m     cr  —  iiw  2 

49.  When  m  =  -  and  n  =  ^  find  value  of  ^>^' +  mn  -  2  7i^^ 

50.  Ifa  =  ^^and6  =  ^^,findvalueof  ^i^  +  1      ^~^ 


iC  +  l 


x-\ 


a  —  h 


a  +  h 


x-\- 


51.   If  a;  =  a  —  2,  find  value  of 


x^l 


-1 


a;  + 


+  1 


52.    If  a  =  m  and  6  =  w,  show  that 


x-1 

a  +  h     a  —  h       4  mn 


a  —  h     a  +  h     w?  —  r^ 


53.   If  a  =  -T-^  and  c  =  .  "^^,  find  value  of  a  in  terms  of  m 


1— c  1— m 

and  reduce  to  simplest  form. 

64.  If  a;  =  -^  and  y  =  -^;  show  that  ^"^ 


2a6 


a-f-6 


a  —  h 


y?  +  f         (x^+^ 


FRACTIONAL    EQUATIONS  56 

FRACTIONAL   EQUATIONS 

I.  NUMERICAL 

Exercise  33 


Solve: 

2  a;     a;     a;     1 

8.    ^--l  +  5-^-  +  2. 

3       5     2     6 

3                      2 

2     ^_?  =  ^  +  ?.  «     4a;  +  l  ,  2a;-l      3^  +  3 

*  3       3       5^5  ^'    '~S~^~~~2~~'~2~' 

3.   1  +  1  =  ^-1.  -^     5a;  +  l  ,  3a;-2_l  +  8a;. 

5     3      15  10.    —^--^-^ — ^— . 

*  3      2~4     3*  „     a;4-l      a;-l_3-a; 


^     2  a;  ,  a;     a;     11     /^ 

6     ^+l4.^±^  =  2. 
•       2^4 


11 


12.    2(a;  +  3)      3(a;  +  l)^o^ 
5  2 


13.    i.(a;  +  2)  =  i(a;-3). 


2a;  +  3      a;-l_o 

11      ^     3  14.   ^(a;-l)-|(a;4-l)  =  0. 

15.  |(a;  +  l)-f(aJ  +  2)=i(a;  +  l). 

16.  (a;  +  l)(a;-i)  =  a^.      . 

17.  |(2a;H-l)  +  2  =  ^(3a;-2). 

18.  ^-i(.  +  3)  =  ^-3. 

19.  a;-J3a;-i(a;-h  1)1=0. 

20.  i[a;-(2a;  +  i)]  =  i(a;  +  i). 


56  FRACTIONAL    EQUATIONS 

23.  2__  3 

24.  fe^V(£=31^=.(5.^-l). 


25.   2i-|(a^  +  3)  =  l|^+l-2^. 
o  o 

26  fe±D-i(^±ll  =  0  31     ^^'-1^3a;  +  l 

3  2  '  *   4a;  +  3     6a;  +  l 

27  -A_  =  _3_.  32     3a;4-2      l^o;-! 
ic  +  l      x-{-2  '       4.x         4     ic  +  1* 

28.  _i-  +  2  =  ^^.  33.    3a^-l_2a;-3 
a;  +  l             x-1  ^x  +  2     Qx-\-5 

29.  ^±1  =  ^^.  34.    _L-+-i-  =  l  ^ 


a;  +  l  a;  +  l      x-l  x^-1 

30.    ?^-l  =  ^Zl3.  35.    ^_+     3  0. 


ic  +  2  a;-2  x-2     x  +  2     3^-4 

36.         3_^— A_  +  i=     9a- 


3a;  +  l      l-3aj  9iB2_-L 

37.    ~i-  =  -J^ ^ 38.    ? +  — ^L_  =  0 

a-'^  +  l      x+1     x'-x  +  l  ar^  +  3a;H-2      (a;  +  2)2 

39     a;^  +  a;4-l^a:^  — a;4-l  .       x 

x  +  1  x-1     "^l-ar^*       . 


40. 


FRACTIONAL    EQUATIONS  67 

X  5  a?^  X 


3a;4-6      6^^-24.         2ic-4 


41.      „      ,^        ,+  2  1 


2a;2  +  a;-l     a!2-x-2     2ar'-5a!  +  2 

43.  ^ l-  =  ^i 1-. 

a;  +  2     x-\-3      .t  +  4     a;-f-5 

44.  -1,+     2  3 


45. 


46. 


47. 


48. 


x-\-l      a^  —  1      1—x 
2  03*^  3         a;  x 


a^  —  1      1  —  x     x-{-l 
3  4.2 


2a;  +  l      4a;2_-i^      l-2x 

xj-3___2_^a^--l 
a;_2     ic2_4     2  +  a;* 

3  2  5 


x-S     l-2a;     2af-7a;4-3 


4a;  +  l  ^      3      _  1  +4a;^ 
x-^2  ~a^-4       2-a;  * 

a;+5      5  —  X     25  —  a^ 

51.   ^ 4 3_^o 

2(2a;  +  5)      2a;=  +  9a;+10     3a;  +  6 

52.  -^-  + 5^ =  ^-+1. 

x-32(a?'-»-6)     a;  +  2 


58  FRACTIONAL    EQUATIONS 

53.         ^       +      ^^     = ^_ 

6a;-6     3-3a;  2a;  +  2* 

64.   2a;-l     4a;-l      .^  ^         -18ar» 


3a;  +  4     6a;-l  21a;-4  +  18ic2 

55.       2     _        3__^        .  1 


a;  +  3     2(aj  +  3)      3(a;  +  3)      4 

_     4a^-2a;4-l  .  4a^  +  2a;  +  l      . 
2x-l       ^       2x^1 

^^    Sx-2  ,  2x-l     X 


58. 


6  2iB  +  3     2 

3aj  +  5     2a;-l      2ic 


18  x-\-5  12 


59  a^     4a;  +  5_2a;  — 1 
3      2a;  +  4~       6 

60  ^^  +  13      6a;-1^3a;  +  2 

8  3  a;- 9         12     ' 

6         9  a;  -  36         6 
62.    ^+^i-    ^-2    -^J-l 


63. 


64. 


66. 


9 

3  a; +  11 

9 

li 

a;-l 

a;-3 

X  - 

-4      a; 

-2 

a;-2 

a;-4 

X  - 

-3     X 

-1 

2a;-3 

3a;- 

1 

x-^1 

2 

4 

5 
4 

1 

5 

3 

3 

2  a; -3 

3a;- 

J._ 

a;  4-1 

1 

=  — 

^BACnonAL  EQUATIONS  6^ 


II.  LITERAL 

Exercise  34 

Solve: 

1.  3a;  +  5a  =  ic  +  8a.  6.  ^ax  =  h{x-\-a). 

2.  2a'4-4a  =  3a;  +  3a.  7.  2(x^a)=^{x-a). 

3.  3x  +  2a  =  4a;  +  a.  8.  (x  +  a)^  =  (a;  —  a)^  +  4 a. 

4.  3aa;  +  4a  =  aa;  +  10a.  9.  (a;  —  a)^  =  (a;  —  6/ +  al 

5.  3aa;  =  a(a;  +  a).  10.  {a  —  h)x  +  {a  +  h)x  =  a?. 

11.  (a  +  6)a;  +  (a  — &)ic  =  «^  +  ca;. 

12.  (aj  +  a)  (a;  —  m)  =  (a;  —  a)  (a;  4- m). 

13.  mx  (x  +  m)  +  7ia;  (a;  —  m)  =  (m  +  ?i)  ar'  +  w  —  n. 

14.  a6  — (a  — 2  6)a;=(2a  — 6)a;  — 1. 

15.  (x  +  2a+by-(x-h2ay  =  b(Sa-^5b). 

16.    ^  +  ^  =  3. 

a     2a 

17     ^_?^  =  1.   ^ 
*   a     3a     2 

18.    'i^'  +  !^  =  m^  +  n^ 


19.    i  +  l  =  l_l. 
m      a?      71      aj 


20.    ca;  +  a  +  -  =  -- 
c     a 


2\a       J      S\a 


1  . 


22. 

x-\-m     3 
a;  — 71      4 

23. 

4              3 

m  +  a;     tti  —  a. 

24. 

aj  +  d^      g 
a;  +  c2      d 

25. 

771  +  1      7/1  +  a; 

771  —  1        m  —  X 

26. 

n(a  —  x)             t\ 

m  —  -^ ^  —  a  =  0. 

2a-a; 

OT 

9                     5 

m4-7i  —  a;     77i  —  Te  +  a; 


60  FRACTIONAL  EQUATIONS 


28.    nk+Jl^^LlzJl.  30. 


X  —  1         X  -\-l 


'"  ^  31. 


««7. 

m     m- 

-n     m-\-n 
30       a;  +  n    _ 

a;  +  2m 

33. 

2a;-a 
X  —  a 

34. 

2x-a 
x+b 

m-\-x 

0?  —  6 

35. 

2 

m  —  x 

2 

1^ 

36. 

1-^ 

1+^ 

1+^ 

m 

1-5 

m 

m 

37. 

m-f  1 

m  — 1 

2     ~ 

'     4 

m  +  a; 

a? 

m  —  x 

m^  —  x^ 

ab  -{-X 

^ 

ab  —  X     y?  —  a^l^ 
3m 


3a;  +  4mH-2n 


x-{-l 


41.  t^Lzl^  =  4^Hhl^. 
i«H-faJ     -ia-fa; 

42.  a;  +  a_.a^  +  «  +  l. 
ic  — a     a;  — a  — 2 


43.    ^  —  2'^     m  +  4a;     ^^p 
m+a;         m  —  x 


.       x  +  a _x  4-  c 
a;  +  c     a;  +  a 


45.   E±^_^LL^  =  o. 
a;  — 5      a  — 6 

c+2aj  4a^ 


46. 


c  — 2a;         4a;^  — c^ 
x 


m 


m  —  n      .      m 
47. 1=  — 

OQ     ^x  —  a_o     b  —  x  m-\ ^—  ^ 

5a;-a^a-10a;  43.       ^ !L_  =  '?Lz: 

•    2x-a       a-4a;*  *    a;-m     a;-n  x 


.f.         X       .       a  nx  ._     2a;— m    x-\-2m  ,  5m     ^ 

40. 1 = •       49. ' 1 =  U. 

m  +  n     n  —  m     w^-v?  2x-\-m    x—2m      x 


SIMULTANEOUS  EQUATIONS  61 

SIMULTANEOUS   EQUATIONS 

I.  NUMERICAL 

Exercise  35 


Solve: 

1.   5x-{-    y  =  ll, 

5. 

x  +  3y  =  5, 

9. 

5x  —  y==16, 

3x  +  2y  =  l. 

3x-{-4:y  =  0. 

x  =  y. 

2.      x-{-2y  =  S, 

6. 

x-4.y  =  7, 

10. 

a;  +  82/  =  -20, 

3x-    y  =  3. 

4.x-    y  =  13. 

3  aj  +  4  2/  =  0. 

3.    2x-\-3y  =  16, 

7. 

2x-13y  =  l, 

11. 

aJ-2^=0, 

3x  +  2y  =  U. 

3x-21y  =  <d. 

4a;-5y-h2=0. 

4.      x-\-2y  =  3, 

8. 

2x  +  3y  =  4, 

12. 

4a;-    2/  =  10, 

3x~    y  =  16. 

6x-    y  =  l. 

72/-2x  =  12. 

13.   y  —  x  —  l  = 

3, 

15.    5 

x-3 

:2/-72  =  5y, 

x—5=-y 

X 

-1  = 

=  15y. 

14.   3x  =  -2y, 

16.   5 

x  +  3 

2/  =  102, 

x  =  35-{-lly. 

7 

2/  +  3 

ia;  =  104. 

-  M=^' 

19. 

1-!=^' 

21. 

x^y 
2     3' 

M=^- 

|  +  .  =  9. 

i+i-»- 

-  M=-^' 

20. 

X     y_7 
3     4     4' 

22. 

hh-^ 

X     y         5     . 
3     2~     6* 

5     3         5 

- 

3     2     9 

62  SIMULTANEOUS  EQUATIONS 

''■l+t''    -1-1=^'      -T-H' 

3     4     4  2^3  2      3  ~       ^' 

26.    ^  +  f  =  -3A,  32.    .-^  =  2  +  2(.-,), 


33. 


2x      y 
3        3 

1 
3' 

14     ^ 

a; 

=  7' 

32/     a^ 
5       8 

-k- 

2a;4-3 

5y-S 

-2, 

7 

4 

i(142/- 

ox        160^- 

23 

*       2 

3     -^' 

3a;  +  l 

.2/  +  l_i 

4 

3 

a;     y 

.3(0^  +  2/4-1) 

3     2~ 

4 

!-«= 

=  1  +  12, 

aj+2/ 
5 

35-2^ 

a; 

a; 
"3* 

x  +  Sy 
3 

-2     42/- 

—  x  +  5 
5'      ' 

3a;-2/ 

+  7     2a; 

+  32/ 

'  +  1 

10 

12 

5 

2 

34. 


35. -, 

4a;+82/+l      7a:-22/-l 

2x+4:y     5x-^6y     Ay 


se.  ^-^t.=(2.-|) 


<?o     6  2/-1  ,  3a;  +  l_5 
30.    — - —  +  — ^ — -^, 

3^+1^14 
5a;-l     2/  +  3  ,  I^a  -22/       5* 

4  7     "^3       ' 

a;+2     y-2     Sjy^Ax)       ^^-   x-y'^' 
^^'   ~S 2-  =  — 4 ' 


y     X     X     y 
4     2     5     3_41 
2       ^3       ^^^     -'  3  5     ~60' 


^_5=^-3(2/-a.) 


SIMULTANEOUS  EQUATIONS  63 

38.  -^_=_-^_, 

4a;-3(a;-?/)_^  .,       x-2(x-\-y) 

39.  (a;  +  3)(2/-l)  =  (aj-3)(2/  +  l), 

a^-^  +  i^  =  (.-3)(,  +  l)  +  llf 

.^    4aj-3  .    2x-3       Qx-l 
40. 


2 

r 

Sx-2y 

"      3     ' 

52/- 

1 

x  +  1 

15^-10 

3 

3x-y 

9 

2x 
3 

Sy 

5 

x-h2y 
4 

=3--7 

6y 

> 

1+^ 

,_S 

\x-y 
5 

-^+t 

41. 


2a5-2/  +  3     05-2^  +  3 

3a;-4y  +  3_        4a;-2y  — 9 
4  ~^  3  ' 


43. 

(K      2/ 

45. 

5+5=2, 

a;     y 

47. 

X     y 

a;     y 

a;      y 

i-l=o. 

a;     2/ 

44. 

13     5 

25     2/     2' 

46. 

3       2         31 

a;       2/         40' 

48. 

1        1  _       5 
2x^Sy         12' 

?  +  l-  =  -2. 

a;     y 

5_10_11^ 
X      y       S' 

111 
3a;     2y      12* 

64  SIMULTANEOUS  EQUATIONS 

49.  ^  +  ^=1,        51.    A  +  X  =  _2,    53.    1^-A  =  _A, 
2x     Sy       '  2x     3y  '  3x     2y         36' 

X        2/6  4:X     Qy       '  ■  8x62/     24 

50.  A  +  A  =  _??,   52.    ?_5  =  _4,       54.    Ah_A  =  8, 
3a;     22/  3  a;     2/  2a;     42/       ' 

2a;     32/  3*  x     y  ^*  3x      2y  3* 

4a;     dy  3a;     22/ 

^ ?_  =  22  -62/  +  4a;  =  26a;2/. 

3a;     32/       ^' 

3a;-5  ,    y-1  /3a;-l\  ,  ^ 

— 2 — +      3"  ^^-y-  2/f— 3 — \  =  xy-x-\-6. 


II.  LITERAL 

Exercise  36 
Solve: 

1.  a;  +  32/  =  7a,  5.         x-\-y  =  m, 
5a;  — 22/  =  18  a.  2a;— 32/  =  w. 

2.  3x-\-2y  =  5a,  6.         a;  +  2/  =  '^  +  ^> 
5a;-f32/  =  8a.  3a;  — 22/  =  m  — ?j.    ^ 

3.  aa;  +  62/  =  l>  7.         x  —  y  =  m  —  n, 
ax  —  6y  =  3.  nx-\-  my  =  2  mn. 

4.  2aa;  +  3&?/=l,  8.   ax-\-hy  —ntj 
3 ax-]- 2 by  =  2.  ex  -{-  dy  =  n 


SIMULTANEOUS  EQUATIONS  65 

9.    x-{-my  =  -l,  ^^     (7n  +  n)x-\-(m-n)y^^ 

y  =  n{x-\-l).  '  m'^  +  7i^ 

10.  mx  =  ny, 

x-\-y  =  a. 

11.  {c-{-d)x=(c  —  d)y, 

x  —  a  =  y. 

x  —  y  =  0. 

13.  (711  +  n)  X -{- cy  =  1, 
ex  -\-  (m-\-  n)  2/  =  1. 

14.  ^  +  ^  =  c, 

a;     2/ 

X     y 
X  y     _  3cd  —  c^ 

'    c-\-d     c  —  d       c^  —  d^' 
x  +  y  =  c. 

16.    i^  +  -  =  a  +  &, 
ox     ay 

X     y 

^    •    y    ^    ^        25.  (g  +  ^>-(^-^)y-i^ 


ma;  —  ny  =  m'-^  +  w^- 

20. 

(a  +  6)x+(a-6)2 

21. 

a            c 

x-\-l     7/H-l      ^ 

n              —  J- 
c             a 

22. 

aU-^a-,-^^ 

ah 

23. 

x-\-y  ,  x-y     ^ 

i»-?/      «  +  ?/_n 

m            n 

24. 

2'  +  »-«  =  3a. 

17.    -^  +  - 


4ccZ 


X       .      ^ ^       2  (£zi^'  =  l.. 


+ 


m  —  n     m-\-n     m^  —  n^  (c  +  c?)2/ 

18  ^+^^^-^«  26.  ^-^  I  y-'^=i 

'   y  +  a     y—2a  '  p—m     p  —  7i 

x  —  a __ x-\-Sa  x  +  m  .  y—  m _ m 

y—a       y+a '  p         m—n     p 

R.   &  S.   EX.  IN  ALG. —  5 


66  SIMULTANEOUS  EQUATIONS 

III.    THREE  OR  MORE  UNKNOWN  QUANTITIES 

Zizercise  37 

Solve : 

1.  Zx-\-    y—2z  =  l,  9.   x-\-y  =  2a, 
2x-3y-{-    z  =  -l,  x  +  z=3a, 

4:X—2y-\-3z  =  14:.  ,          ^ 

^  y  +  z  =2a. 

2.  x  +  3y+    2  =  1, 

2x+    y-3z  =  l,  ^^'   ^  +  2y+    z  =  a, 

3x-^2y-2z  =  -2.  x+    y-^2z  =  b, 

3.  2x  +  3y-5z  =  0,  2/+  z  +  2x  =  c. 
3x-4:y-2z  =  -3, 

2y-3x  +  Sz  =  7.  '^'   o^  +  20  =  2(, - .), 

4.  2.  +  3,  +  4.  =  12,  y  +  20  =  3iz-x)^ 
3a.-42/  +  5.  =  2,  .  +  20  =  2(..  -  2,). 
4:X  +  5y-\-6z==24..  12.   x  +  y-^z  =  a-\-h, 

5.  3a;-  y-{-2z  =  -ll,  x-^y-z  =  a-b, 
3y  +  2x-  .  =  -12,  2/  +  ^-^  =  c-a. 
32!+    x  +  2y  =  -20. 

6.  2a;4-2/-10;2  =  20,  13     ^4.^  +  ^  =  ^ 

^     ,  Q        1K  3^2^4     12* 
—  y — oz-\-3x  =  lbj 

^  4^3     2     12' 


7.    x  +  y  =  z  +  3y 


X     y  .  z  _  5 


y  =  Sx-8,  2~4  +  3     2 
2  —  a?  =  4. 

8.   x  +  2y  =  25,  ^^'   1^;  + J2/  + J.  =  23, 

2/-22;  =  0,  icc  +  iy  +  i2  =  28, 

a;  +  30  =  2O.  ix-\-iy  +  ^z  =  27. 


SIMULTANEOUS  EQUATIONS  67 


''•    2  +  2/     5' 

S+z      f 

z         2 
4  +  a;     3' 

16.   ay  +  hx  =  l, 

ex  +  az=  1, 

bz  +  cy=l. 

"•  A+ft-^ 

X 

x  +  y-\-u  =  SAf 
2/  +  z  +  w  =  36. 


20. 

1.2     1_^ 

X     y     z 

2_4  +  3^_3_ 

X     y     z 

?_l+?=i 

X     y     z     2' 

21. 

_  +  -  =  w, 
a?     2/ 

a;     z 

2,     z 

2|, 


2*     32/      2 

1+1+ i=r.  22.  i5+§_§=4, 

X     y     2z  X      y     z 

9      4     4. 

18.   a;  +  2/  +  2;  =  33,  -"7,  +  ^"=^' 


X      y     z 


+  z+u  =  S5,  . 

?  +5-1  =  2. 


352/     - 


.o     1^1      ^  23.    ^^  =  7i, 

19.   _  +  _  =  -,  x  —  y 
X     y     Q 

1_^1_7  -^==li 
^  +  ^"12' 

1  +  1  =  1  -^^=^i- 


a;      2 


4  2/-^ 


68  SIMULTANEOUS  EQUATIONS 


24.    "  +  -^  +  5  =  1 

X     y     z     2 

25. 

mx  +  ny  =  a(m-{-  n), 
mx  -\-az  =  n{a-\-  m), 

a  .  b     c     1 

26. 

ny  -\-az  =  m(a  +  n). 
x  +  y=22, 

^_^_£— _ 

1 

2/  +  ^  =  18, 

X     y     z 

2 

• 

2;  -j-  W  =  14, 
W-\-U  =  10f 

w  +  a?  =  16. 

IV.    PROBLEMS  IN   SIMULTANEOUS  EQUATIONS 
Exercise  38 

1.  A  man  purchased  20  acres  of  land  for  $1640.  Part  of 
it  was  bought  for  $  90  an  acre  and  the  remainder  for  $  50  an 
acre.     How  many  acres  were  there  in  each  portion  ? 

2.  A  man  and  a  boy  together  weigh  230  pounds,  and  twice 
the  man's  weight  is  60  pounds  more  than  3  times  the  boy's 
weight.     Find  the  weight  of  each. 

3.  Three  horses  and  4  cows  can  be  bought  for  $610,  but 
at  the  same  rates  it  takes  $720  to  purchase  4  horses  and  3 
cows.     Find  the  price  of  each  per  head. 

4.  If  half  of  A's  money  is  added  to  B's  money,  the  sum  is 
$  170 ;  but  if  half  of  B's  is  added  to  A's  money,  the  sum  is 
$  160.     How  much  money  has  each  ? 

5.  In  10  hours  A  walks  1  mile  more  than  B  walks  in  8 
hours.  In  5  hours  B  walks  5^  miles  less  than  A  walks  in  7 
hours.     How  many  miles  does  each  walk  per  hour  ? 

6.  If  the  numerator  of  a  certain  fraction  is  subtracted  from 
the  denominator,  the  remainder  is  21 ;  but  if  the  denominator 
is  subtracted  from  8  times  the  numerator,  the  remainder  is  —  7^ 
Find  the  fraction, 


SIMULTANEOUS   EQUATIONS  69 

7.  In  a  certain  town  meeting  312  voters  were  present,  and 
a  motion  was  carried  by  a  majority  of  8  votes.  How  many 
voted  for  and  against  the  motion  ? 

8.  Two  men  had  together  $  100,  and  if  the  first  had  given 
$  10  to  the  second,  each  would  then  have  had  the  same  amount. 
How  much  had  each  originally  ? 

9.  If  2  is  added  to  both  numerator  and  denominator  of  a 
certain  fraction,  the  resulting  fraction  is  |.  If  1  is  subtracted 
from  both  numerator  and  denominator,  the  new  fraction  is  ^. 
Find  the  original  fraction. 

10.  A  boatman  can  row  20  miles  down  a  stream  and  back  in 
10  hours,  the  current  being  uniform.  He  can  row  2  miles  up- 
stream in  the  same  time  that  he  can  row  3  miles  downstream. 
Find  his  rate  per  hour  both  down  and  upstream. 

11.  If  the  width  of  a  field  were  increased  1  rod  and  the 
length  2  rods,  the  area  would  be  34  square  rods  greater ;  but  if 
the  width  were  decreased  2  rods  and  the  length  increased  3 
rods,  the  area  would  be  unchanged.  Find  the  length  and  width 
of  the  field. 

12.  Find  a  fraction  such  that  if  you  double  the  numerator 
and  add  3  to  the  denominator  the  result  is  f ;  but  if  you  add  3 
to  the  numerator  and  double  the  denominator  the  result  is  |^. 

13.  Two  sheep-owners  met.  A  said,  "  Sell  me  4  of  your 
sheep,  and  I  shall  have  twice  as  many  as  you  have."  B  said, 
"No;  sell  me  1  of  yours,  and  we  shall  each  have  the  same 
number."     How  many  had  each  ? 

14.  When  a  certain  number  of  two  digits  is  doubled  and  in- 
creased by  4,  the  result  is  the  same  as  if  the  digits  had  been 
reversed  and  this  number  decreased  by  22.  The  number  is  2 
less  than  3  times  the  sum  of  its  digits.     Find  the  number. 


70  SIMULTANEOUS  EQUATIONS 

15.  If  I  divide  a  certain  number  by  3  more  than  the  sum  of 
its  two  digits,  I  get  a  quotient  of  3  and  a  remainder  of  8.  But 
if  I  reverse  the  order  of  the  digits  and  divide  by  twice  the 
sum  of  the  digits  in  the  same  reversed  order,  my  quotient  is 
3  and  remainder  11.     Mnd  the  number. 

16.  A  boy  bought  5  apples  and  3  oranges  for  25  cents,  4 
oranges  and  5  pears  for  35  cents,  2  pears  and  7  apples  for  20 
cents.     Find  the  prices  paid  for  each  apple,  orange,  and  pear. 

17.  Find  3  numbers  such  that  if  each  be  added  to  |  the  sum 
of  the  others,  the  results  will  be  32,  28,  and  30  respectively. 

18.  The  sum  of  the  three  digits  of  a  number  is  12.  The 
hundreds'  digit  is  one  half  the  sum  of  the  other  two,  and  the 
units'  digit  is  ^  the  number  composed  of  the  other  two  in 
the  original  order.     Find  the  number. 

19.  Three  boys  together  weigh  300  pounds.  Half  the  sum 
of  the  weights  of  the  first  and  the  third  equals  the  weight  of 
the  second.  The  sum  of  the  weights  of  the  second  and  the 
third  divided  by  the  difference  between  the  weights  of  the  third 
and  the  first  gives  a  quotient  of  5  and  a  remainder  of  20. 
Find  the  weight  of  each. 

20.  A  and  B  together  can  do  a  certain  piece  of  work  in  3 
days,  A  and  C  the  same  work  in  4  days,  B  and  C  the  same 
work  in  6  days.  How  long  will  it  take  each  alone  to  do  the 
work  ?     How  long  will  it  take  all  working  together  ? 

21.  Some  books  were  divided  among  3  boys,  so  that  the  first 
had  12  less  than  half  of  all,  the  second  1  less  than  half  the 
remainder,  and  the  third  17.     Find  the  number  each  received. 

22.  A  boy  has  100  pieces  of  silver.  The  value  of  the  quar- 
ters is  3  times  the  value  of  the  dimes,  and  the  sum  of  the 
values  of  the  half  dollars  and  dimes  divided  by  the  difference 
of  the  values  of  the  quarters  and  half  dollars  is  f.  Find  the 
number  of  dimes,  quarters,  and  half  dollars. 


INVOLUTION  AND  EVOLUTION  71 

INVOLUTION   AND   EVOLUTION 

L    MONOMIALS 

Exercise  39 


Write  the  value  of : 

1.  (2a)l            9.    {-2mn)\  ^^  (Sa^\         ^1.  p^^V. 

2.  (3  ay.         10.    (-2  ay.  ^      ^  ^ 

^-    ^^'*^'-           12.    (^m'ny)\    ^g.  ^-^^^Y.  23.  ^^^'^'^* 


/     2  amV 

V    3cdy 


®-    ^~-^")-      13.    aaO^ 
7.    (Say.  '    \S 


20.    -f-^X  25.    r^^^Y. 


40. 


^S2a'' 


8.  (-2a2)^  15.    (-^aby. 

26.  V4m^  34.    -y/lQ  m*n\ 

27.  Vl6mV.  35.    ^Wm^n"^. 

28.  ^8?d^.  3e.    ^/4^.  ^1-    \'^K^J 

29.  ^27?. 


30.    V64mV. 


^ 


42.    A^/-32'''''^''" 


31.  ^327^.  ^^     ^/25^  43.    V-27(a  +  6)^. 

32.  VlOOa^y.  44.    ^Sl  (m-ny. 


33.    ■v/-64a^c^  '    >'343ci2  45.    V-32(a  +  l)' 


72  INVOLUTION  AND  EVOLUTION 

n.    INVOLUTION  — BINOMIALS 
Exercise  40 


Expand : 

1. 

(a  +  by. 

12. 

(d^- 

-4  c)*. 

20. 

(- 

-!)• 

2. 

(a  +  by. 

13. 

(d^- 

-3  c)'. 

3. 

(a  +  m)^ 

14. 

(c^-\-2cy. 

21. 

(aH 

-a^-l)^ 

4. 
5. 

(a  +  2y 

(a-sy 

15. 

22. 

i^- 

3  a; +  2/. 

6. 

(2a  +  3y. 

16. 

(2  a 

-I)'- 

23. 

(m*- 

-7l2_3^^\ 

7. 

iSa-2y, 

17. 

fab       \' 

24. 

K  + 

■  a  +  1)^ 

8. 

(2a2  +  5)^ 

25. 

(«=- 

■a  +  lf- 

9. 

(ab  -  ly 

18. 

V2"' 

-')■ 

26. 

(a  + 

c-i)'- 

10. 
11. 

(a-2y. 
(a'b'-{-2cy. 

19. 

(i- 

-11- 

27. 

(2(1- 

-a'-\-iy 

HI.    EVOLUTION  — SQUARE  ROOT 
Exercise  41 

Extract  the  square  root  of : 

1.  aj*  +  2a^  +  5aj2-t-4aj  +  4. 

2.  x*-6x--{-17x'-24:X-\-16. 

3.  aj«  +  2ar^  +  aj'-2a^-2a;2  +  l. 

4.  a;^_2a;3  +  lla^-10a;  +  25. 

5.  a^  +  4:a;*-8ar'  +  4ar^-16a;  +  16. 

6.  4a;*-20x3_,_37  3^_3Q^_,_9 


INVOLUTION  AND  EVOLUTION  73 

7.  ^-12x-2x'-\-4.s?-\-Q^. 

8.  a^  —  2  a^x  +  5  aV  —  6  aV  +  6  aV  —  4  aa^  +  a;«. 

9.  12a;3-30a;  +  4a;^4-25-llar^. 

10.  25  a;2y  _^  2  a;y  H- x«  -  8  ary- 12  a^?/^  + 36^^- 4  ar*?/. 

11.  ar'-2ar^-14a;  +  49  +  14x^4-a^. 

12.  -  +  X^+    3    +T+9 

13     ^'_i^4.??^  +  a;  +  i.       . 
^^'    9        3^6    ^"^^16 

14.  ^  +  i^+2-i^  +  4 

15.  4  +  4a-a^  +  -+  — --  +  -. 

16.  f  +  4  +  6_i_2a. 
9      Of     a  3 

17.  c«-|  +  ^*-f-^'  +  <^  +  f^--  +  Ti-,- 

2       16        2a  4aa4a^ 

Extract,  to  three  terms,  the  square  root  of : 

18.  1  4- 3a.  20.    a^  +  9  &. 

19.  l-5a.  21.   a^+4a;. 

Extract  the  fourth  root  of : 

22.  81a:4-216a^  +  216a^-96a;  +  16. 

23.  a^-12a362_^54a26*-108a6«  +  8168. 

24.  16a^  +  16a»a;  +  6a2a^  +  aa^  +  — . 

■         16 


74 


INVOLUTION  AND   EVOLUTION 


IV.   EVOLUTION  — CUBE  ROOT 
Exercise  42 

Extract  the  cube  root  of : 

1.  a3  +  9a2  +  27a  +  27. 

2.  a«-15a^  +  75a^-125a» 

3.  a«  +  6a^  +  15a^  +  20a3  +  15a2  +  6a  +  l. 

4.  1  -  9  a  +  33  a^  -  63  a^  4-  66  a^  -  36  a^  +  8  a«. 

5.  60  a^  4- 1  +  240  a;^  +  64  cc«  -  192  ar^  -  160  aj^  -  12  jb. 

6.  m®  — 3m*  +  5m^  — 3  m  — 1. 

7.  18  a*  +  90  a^  +  125  -  3  a«  -  31  a^  -  75  a  +  a\ 

8.  m3  +  m2+^+  ^ 


9.   a«  +  ^V^^''^' 


a^ 


3      27 

4- 


a^ 


10.    a^-.^"'^  ■  ^^^'      ^' 


2c   "*"  4c2 


8c^ 


11.  a^-3a:5  +  2a^-a:3^2^_«_l.. 

3       3     27 
Extract  the  sixth  root  of : 

12.  1  -  6  ?ri  +  15  m2  -  20  m^  +  15  m<  -  6  m«  +  m«. 

13.  a«  +  60  a^62  ^240  a^b*  +  64  6« - 12  a'b  - 160  a^b^- 192  ab^ 


V.  EVOLUTION— NUMERICAL 
Exercise  43 
Find  the  square  root  of : 

1.  3969.         5.  15129. 

2.  6561.         6.  93636. 

3.  8464.         7.  1772.41. 

4.  10404.        8i  2672.89. 


9.  .986049. 

10.  .01449616. 

11.  .01018081. 

12.  .000104101209. 


INVOLUTION  AND  EVOLUTION 


16 


Find,  to  four  decimal  places,  the  square  root  of: 

13.  8.  16.    2.5. 

14.  14.  17.   37.561.  20.   1.0405. 

15.  175.  18.    .375,  21.    .0035. 


19.   |. 


Find  the  cube  root  of : 

22.  42875.  25.  12977875.  28.  .001481544. 

23.  250047.  26.  28652616.  29.  34328.125. 

24.  1860867.  27.  74.618461.  30.  20.570824. 

Find,  to  two  decimal  places,  the  cube  root  of: 

31.  9.  33.  7.3.  35.  |. 

32.  67.  34.  2J.  36.  .0042. 


Find,  to  two  decimal  places,  the  value  of : 
37.    V5.  38.    Vi5.  39.    ViO. 


41.  V^. 

42.  v:oo7. 

43.  VIl2. 

44.  a/IO. 

45.  V.0017. 


46.  V2V2. 

47.  V5  -f  V3. 


48.  V5-f-V3. 

49.  ^10-fV7. 

50.  Vl5-3^i8: 


51. 


40.    V.9. 

^5  +  V5 


52.    J10+4V3. 
^       V2 


53.    V.038  4-V.009. 

54.  Vv:5+\/A 


55.  vio+Vio+^10. 


REVIEW 

Exercise  44 

1.  Find  the  H.  C.  F.  of  5  a*  -  4  a^  -  64  and  a' +  a^  -  20. 

2.  Take  ic^+3  from  a^  —  2.x^-^Xj  and  multiply   the   re- 
mainder by  a;  (a?  +  3). 

3.  Find  four  terms  of  a^  -^  (a  —  2). 

4.  Prove  that  (''+^)'-<"-^)'  =  a6. 

4 


5.  Find  the  value  of  Va^  +  2/^  +  2;^  —  (ic  —  2/  —  zy,  when 
a;  =  3,  2/  =  —  3,  and  z  =  0. 

6.  What  is  the  remainder  if  (a  —  2) (a  —  3) (a-  —  a  +  5)  is 
divided  by  (a  -  l)(a  +  2)  ? 

7.  A  certain  divisor  is  a^  -{-x  —  2  and  the  corresponding 
quotient  is  ay^  —  x—  1.     Find  the  corresponding  dividend. 

8.  What  quotient  will  result  if  the  sum  of  x^  —  5x^  -{-Sx 
and  2qi^  —  5x  —  1  is  divided  by  the  sum  of  3 i»^  —  2 a;  —  3  and 
-2i»2-h4? 

9.  Solve    ^^-^  +  ^-=^=2. 

x-\-2       X  —  5 

10.    Prove  that 

("-fe?)^fe?)+(4ef)'=-- 

76 


REVIEW  77 


11.    If  m= r,   ^  = n^  P  — 


a  +  1'          a  +  2'^     a  +  3 
find  the  value  of       z hi h 


1  —  m     1  —  n     1— i> 

12.  What  value  of  x  will  make  the  expression 

3(a;  +  2)-4(a;-3) 
equal  to  twice  the  value  of  a;  ? 

13.  If  -Ti =  9,  find  the  value  of  — ' 

Sa—x  x—a 

14.  Show  that 


5a; 


-2a:-[4-i(a.-  +  4)-21a;-3-(a;  +  2)S-4]-^  =  a 

15.  Show  that 

a(h-\-c  —  a)        h(c-\-a  —  h)        c(a-{-b  —  c)  _^ 
(a  _  6)  (c  -  a)  "•"  (6  -  c)  (a  -  6)  "^  (c  -  a)  (b  -  c)  ~    ' 

a-{-4:b     2     46  —  g 

..    .Q-       r^     a-46  46  +  a      /2       IN 

16.  Simplify  -^— X^--2^} 

17     Ifg-     2findvalucof(^~^^^^~^^>      (l+a)(l+2a) 

17.  It  a-     ^tmdvalueot jf^^^ (r=2^) 

18.  Solve  ^-^^-1-^  +  ^  +  ^=0. 

2g  — a;  a;  +  2a 

19.  Simplify  


6  («-!)(« -2) 

a  — 1  a  +  l 


.O.Add4[^-l(ao-l)]a„<i3[^_l(c  +  l)-2]. 


78  REVIEW 


21.    Solve  — ; ; —  =  m-\-n. 

x-\-n     x-\-m 


22.  Show  that  (a-\-b)(a+b-l)=a(a-'l)-\-2ab-\-h(b-l). 

23.  Find  the  value  of  — — a(x-^l) ^^ ^ 

IX  X 


when  X 


24.    If  a  =  ^^^  and  c=^^,  findcwhen  6  =  -l. 
62  a  4-  3 

26.    Simplify  a-lb-o-\2a-2,-U3o-m, 

a— J 

when  a  =  1,  5  =  2,  c  =  3. 

4a;     .4 

3i»  +  l     T  3       5x  +  2 

26.    Solve  ic- 


5  2  10     11  a; +  6 

27.  What  must  be  the  value  of  m  in  order  that 

6a^- a^- 11  a^-lOa-m 
may  be  exactly  divisible  by  2 a^  —  3a  —  1? 

28.  Show  that  ^(^-'^)_^JP±^^a  when  a  =  x-c. 


29.  Given  that  m  =  - r  and  ti  =  — - — -, 

1  +  a^  1  +  a^ 

prove  that  m^  +  n^  =  l. 

30.  Prove  that  —  satisfies  the  equation 


3 


4a;  — m     x-^m__^ 
2x  —  m     x^m~ 


REVIEW  79 

31.  Solve  (a;-f3)(2/-2)  =  (a;-5)(2/  +  4)  +  16, 

{x  —  y){x  +  l)  =  l  —  x{y-x). 

32.  If  x  =  ^^  and  a  =  ^~^^  find  x  in  terms  of  m. 

33.  Solve  a(a  —  x)  =  b(x-^y  —  a), 

a(j/  —  b  —  x)  =  b(y  —  b). 

34.  When  x  =  2,  find  the  value  of 

"-[-K''-^i)-|f-i)-<-8}-f]- 

35.  Solve  (a-26)aj  +  a2  +  52  =  (2a-&)a;-2a6. 

36.  What  must  be  added  to  2[l-3a;fl-2a;(l-5a;)j]  to 
produce  -5-3x{l-2xy? 

37.  li  x  =  — ^^  and  m  =  ^  T"    ,  find  value  of  a;  in  terms  of  n. 

w  2    ' 

38.  What  must  be  the  value  of  m  in  order  that  x  =  2  may- 
be the  solution  of  m{x  —  m)—S(x  —  S)? 

39.  Arrange  (m  —  2)*  —  (m  —  1)^  +  3  (2  —  m)^  in  ascending 
powers  of  m. 

40.  Solve  ^-^=  ^  +  ^. 

a;  +  2?i     a;  — 2m 

41.  Expand  [m- 3 n-f3(m-n) -n}]*. 

42.  Find  the  square  root  of 

(m  -  nyi(m  -ny-2  (m^  +  n^)]  +  2  (m^  +  n^. 

43.  Solve  ^-:=i^  +  ^-ii^  +  ^:::i-^  =  l  +  ?  +  ?. 

en  cm         mn       m     n     c 


80  REVIEW 

44.  What  expression  multiplied  by  itself  will  give  tlie  ex- 
pression    4^rc6_i2a^  +  5x'-hl4.a^-llx'-4:X-{-4.? 

45.  Expand  (a^  —  8cy  and  extract  square  root  of  the  result. 

46.  Divide  1 ^^„  by  1 — —^ — —  and  extract  square 

(a-\-by    -^        d'-ab  +  y' 

root  of  the  quotient. 


47.  Simplify  V(a-2c)«. 

48.  Show  that  the  difference  between  the  squares  of  any 
two  consecutive  numbers  is  1  more  than  double  the  smaller 
number.  ^ 


49.    Prove  that  q r is  a  perfect  square. 


a     a^  —  a-  4-  a 


50.  Compute  ^0.4  -f-  V.004  to  three  decimal  places. 

51.  Find  the  cube  root  of  [2  a  -  (a  +  l-(a-2))J. 

52.  If  a;  =  ^L+1,  find  the  value  of 

a  — 1 


^(x  +  4:){x-2)-2(x  -  4). 


53.  Simplify  \VlO  — V5  to  two  decimal  places. 

--     oi      a  —  3a;     b  —  2x     -, 

54.  Solve  = 1. 

64-a;         a  +  x 

55.  Find  the  square  root  of 

(a^_4a;4-3)(a^-9)(a^  +  2a;-3) 
x^-^ex-{-9 

56.  Find,  to  two  decimal  places :  ^  /   "^         . 

A/  V150 

57.  Solve  15y  —  14:X  =  —  4:xy',  SOy —  IS x  =  17 ocy. 


EXPONENTS  81 


EXPONENTS 

I.  TRANSFORMATIONS 

Exercise  45 

Express  with  fractional  exponents  : 


1. 

2. 

Va. 

<ra. 

2V^. 

5. 
6. 

7. 
8. 

9.' 
10. 
11. 
12. 

3  Va-". 
V4al 

13. 
14. 
15. 
16. 

^27  a^. 

3. 

^/a6Vd*. 

4. 

^32a:^2/'V^ 

Express  with  radical  signs: 


17. 

aK 

22. 

abK 

27. 

ahl 

32. 

2  c^dl 

18. 

aK 

23. 

abc^. 

28. 

(ab)l 

33. 

a^^y. 

19. 

aK 

24. 

3  abhi 

29. 

1 

34. 

2ahhK 

20. 

«l 

25. 

ahl 

30. 

3  c^d^. 

35. 

4  c^d-. 

21. 

5aK 

26. 

ahK 

31. 

7cd\ 

36. 

m    » 

5  a"6«. 

In  the  following,  transfer  to  denominators  all  factors  having 
negative  exponents : 


37. 

ab-\ 

42. 

3  a-^b-\ 

47. 

4  a  ^x. 

38. 

ah-'c-\ 

43. 

a-'bc-\ 

48. 

3-^m. 

39. 

2  abc-\ 

44. 

a'b-'c-K 

49. 

9-kd. 

40. 

abh-\ 

45. 

2-'a-\ 

50. 

-  2-^a-^bc 

41. 

1  a~\ 

46. 

3a'b-K 

51. 

-|a-'6c. 

R.   &  8.   EX.   IN  ALG,  - 

-6 

82 


52 


53.    — 


} 

EXPONENTS 

Write  the 

following  without  denominators : 

a' 
b' 

54.    2^. 

mn 

^6     ^^' 

KG     2  mri 

2k, 

c 

66-    !|- 
cd 

57.      2«   . 

59.     ^«" 

60. 

4.C 

61. 

3a-^ 

a-'x-i  ^  „^. 


Express  the  following  -vrith  positive  exponents : 

62.  3a-\  _3_.  4VFJ 

63.  2  am-*.  ^   ^  «"' 

64.  a'^b-'c.  72     i^.  77     ^~'^~' 

65.  mn~^. 

66.  2a-^6V^  73.   ^^=^-  78 


67.  om~^6"^c. 

68.  ab~^xy-\ 


tA, 

(xZ-^ 

73. 

2a6-^ 

Sm-^n 

74. 

1 

75. 

2-V 

0-^6 


2Va-2c- 


79. 


3  mn~- 
4  a-icd-3 


69.    mn-V^^.  ^_,^  ^^     ^-'ab' 


70.   a-^6-V\  '     a-^  SVa;"^ 

Find  the  numerical  value  of  the  following : 

81.  4i.  85.  16^  89.    125i  93.    4"^. 

82.  9i  86.  27*.  90.    (-27)i  94.    9~^. 

83.  4l  87.  27*.  91.    (-64)1  95.    16"^. 

84.  9I  88.  8li               92.    (-125)1  96.    (-27)"i 
97.    36"*.                     98.    (-32)-^.  99.    -s/^^^. 


EXPONENTS  83 

100.  ^J/(=:27p.  109.    Sl-i.l;.  117.  -(-2i^)-l 

101.  (^=:27/.  jj^     2-3-2--  "^-  ^*)'*- 

102.  ^K      ,,,;  ,6-i".8i    "'■  ^-«^"*- 

103.  (^16)'.       ^^^  gi.  j__      120.  ar^-^(#. 

104.  (<^^/.  '  32*'  121.  3-^.  A. 

105.  2-^.3->.  113-   9-* -81*.  ,-j   „j 

122  T 

106.  3-2.2-2.  114.    2-2.32.4.6-1.  •       27^ 

107.  9-^.27*.  115.    (2J)i  16-^ .  27"^ 

108.  16^.8"*.  116.    (l^^^)"^.  *  9-^.64-* 

124.    (4-3 .  3-" .  23)  -^  (16-^  .  27"*  •  Sl"^). 

Perform  the  indicated  operations  in  the  following : 


125.    w 

« .  a-\ 

130. 

(i-«  •  a». 

135.   m-^-m^. 

126.    0/ 

' .  a-\ 

131. 

a^.aK 

136.    171^ -m^. 

127.    a 

.a-8. 

132. 

a  '  ai 

137.    a-*-i-a\ 

128.    a' 

^ .  a-2. 

133. 

a-i .  a-i 

138.    a-3-^a-2. 

129.    a- 

-' .  a-^ 

134. 

aKa-i. 

139.   a-^^al 

140. 

a^cc .  ax-\ 

147. 

8*4-9i 

141. 

amhr^ .  a^m" 

-V. 

148. 

8-^-9-i 

142. 

2a62.3a-i6- 

-1^ 

149. 

a'h  .  a-162 .  ah-\ 

143. 

a«  +  6«. 

150. 

2  a  .  3  a^  .  a-*. 

144. 

3a«-(3a)«. 

151. 

3  a^a^^/  •  ohcx-\ 

145. 

(a  +  &)«. 

152. 

o?^fx  •  a^ic-^ 

146. 

(-2)-3-(- 

-3)-^ 

153. 

2Va-3aV^-a;"^. 

84 


EXPONENTS 


157. 


168, 


163. 


164. 


165. 


166. 


167. 


168. 


x-Wx 


154.  a^Vx  .  a^Vx^  h-  aV^. 

155.  2a-^V^.3aiVx^^-ax-\ 

156.  x'^aVc^-^x-^a^Vcd^. 


159, 


160. 


VaV 

a^\a^Vx 


161 


162. 


3a;-^ 


a^a; 


'VacVac 


3  m  ^Vo^ 


mn- 


^Vm^ 


Sx'^V^   ■ 

a~^^ax~^ 
xr^-y/a-^x 

2aWx^' 


a'^y/d 


c'Voc^ 


169. 


170. 


171. 


172. 


173. 


174, 


oT^x-' 


2aV9a^ 


3  c-s/21  x--a-' 

4 


'V^TlV^ 


12/- 


V4  a^x- 


Keduce  to  the  simplest  form : 

175.  {aj.  178.    (a3)-2. 

176.  (a2)3.  179.    (a-^f. 

177.  (a2)-i.  180.    (2a'y. 


3x^-s/-Ua-^x 

a/-27V64^ 
-^-125a-«' 

181.  (2a-3)-2. 

182.  (a-2)-^. 

183.  (4a-^i 


(a-^-i 

EX 
198.    ( 

PONENTS 

212. 

85 

184. 

(S/-32c^«)-3. 

185. 

(Sx^)-^ 

199.    ( 

2-'a-^y. 

213. 

'\/-81-*a«. 

186. 

{x-'^)-\ 

200.    { 

;8-^)^. 

214. 

(9a-'y-r^. 

187. 

(5aby. 

201.    ( 

;-8-^)^ 

215. 

(16a-V2/2)-l 

188. 

(a'by. 

202.    ( 

;- 27-^1 

216. 

{aV^'y. 

189. 

(a-'b-'y. 

203.    ( 

;-i25)-^. 

217. 

(a-'V^y. 

190. 

(ab-r^ 

204.    ( 

;-8a2)i 

218. 

(a-WaV<)-^. 

191. 

(a*b'c-^^. 

205.    ( 

'2a-'c^y. 

219. 

(a-Va-^-l 

192. 

(646c-2)-l 

206.    ( 

[2ah-'y)-\ 

220. 

(2a'^V^'y'. 

193. 

(-4a)l 
(-aby. 
(-2a'by. 

207.  ( 

208.  ( 

209.  < 

[2m^np-Y\ 

221. 
222. 
223. 

(ab-Wa-'by^- 

194. 

:^-sa-^y. 

(aV4a-^)-3. 

195. 

[■y/16a'b')-^ 

(x-^yjxVwy. 

196. 

{-2aby, 

210. 

[2a-^Sa-y\ 

224. 

l(Va-'by\\ 

197. 

{-^a)-\ 

211. 

[8a-2c-3)-l 

225. 

l(^Sa*by\\ 

226.  '  l{</27'a-'yy\ 

227.  yjie-'x^^. 

228.  (^[m^)^. 

3/ ; 

229.  y-SVaF^. 

230.  (yjVu^y. 


234.  [(64ar^.^]- 

235.  ^j(16x'y-'Vxy)-^ 


236. 


VV^Vm      -\/- 


mx '  m 


231.  (a-'yJ27arix^)-\ 

232.  l^[(^</^Wr'f. 

233.  (^125  a^V^-^ 


237.  (V27^)-2-- (V9¥-3)-i. 

238.  \m~^n^^mn~^yln^)  ' 


^6  EXPONENTS 


240.   y25a-^bVx-''-r-y9a-'b-^Vs^, 


3 

242. 


243. 


244. 


245.    \xl^^^'\\^^^\~\ 

246.  ^M^;y6     c-3^g^yi^ 


Vc-'  Vad- 


1^-^ 


247.     \\l-     V".  J27«V^|-'. 

.48.    [^^"^^-.g^^^J 

249.    (V     •^^"'^'_.  ^332^i^F^r'. 
I  ^25a36-i^c2  J 

252.    iVa^ft-W^'c-iaVcVWaSc) 


EXPONENTS  87 


Collect; 


253.  (|)i+(^V)i  +  (32)i 

254.  (2)-2  4.8^-4-\ 

255.  3-2-27-^4-9"^. 

256.  (^?yVl6-f  +  A+(_2)-». 

257.  (3)-i_  2-3 +  (1^)^  +  128-1 

258.  9-^  +  13  a;*'  + 1"^  +  {^)-\ 

259.  ■^2TF'5_:^  +  J-_128-*. 

2"^      8"^ 

260.  7aJ»-(7a;)«-17^  +  A. 

2 

261.  8i-*-A-  +  :^-# 

27-^     V34     9-^ 
Simplify : 

262.  2".  22-^2".  271.  (a^+i)  V*)' '  («'"0~^- 

263.  4  "-2  .  82-"  .  2».  m  _n        n 

264.  (4-.  2-)  ^8.  '^'-  (-'^'^O-. 

265.  (af^-^-^i .  (a^-r  .  ar=.  273.  (3^+2  _^  3  •  S'*)  ^  (9  •  3"+^. 

266.  {ahy+y  -^  a'fb".  274.  (a^^-^^  •  a;-^«-*)  -^  a;"'*-^. 

267.  a'"+"  .  a^"*-"  •  a""^. 


275.    S[(a)'»-i]-'-i  ;""•+». 

276.  [K«^-r^r]-[K^m- 


268.  ar^^a^"*. 

269.  [(a'=+^)^-^  -5-  (a^-*)^. 

270.  (x"'-^y(xP-''y(oif-^y.  277.    (Va'^^-j-Va)'^^^^ 


EXPONENTS 


278 


279 


2n+iy2  . 2"-3 


1       "*^  1 

280.       KC'""'0'"{     m2-l.^„»+l^ 


3n 
a+6  g— 6  2a    1 

283.    ([ic  *=  ]  .  [ic~]  -h  a;~^)^ 


284 


-.  [^'{5)•(')(S)r■ 


II.    MISCELLANEOUS  APPLICATIOJSTS  OF  EXPONENTS 

Exercise  46 

Multiply  : 

1.  a-2  -  2  a-^b-^  +  &~'  by  a^  -  b-\ 

2.  a^  +  aM4-2>^  by  a^-{-b\ 

3.  a-2-2a-^  +  3-a  by  3a-^-2-2a. 

4.  a^  — aM  +  6^  by  a^  +  ah^  +  b\ 

5.  a-3-2a-2  +  3a-i  +  l  by  a-2-3a-i-l. 

6.  3a^-6a^  +  4  by  a^  +  2aJ-3. 

7.  sJ-a^-{-2-4.ar^  hy  '2a^-S  +  2ari 

8.  ^-f.l^_-^  +  2-^by  ^-2  +  ^'. 

V^     Va;      Va;  Va        Va;  Va 


EXPONENTS  %% 

Multiply  the  following  by  inspection : ' 


9. 

[a-'  +  iy. 

18. 

(a-2-3)(a-2  +  2). 

10. 

:a-  +  3)l 

19. 

(a^+4)(a^-h5). 

11. 

[a-^-4.y. 

20. 

(5-«-2)(3  +  a-^. 

12.    ( 

[a-'  +  h-y. 

21. 

(a^  -  6^)(a^  4-  6^). 

13. 

[a^  +  b-')\ 

22. 

(a-^-3)(a-^-2). 

14. 

[a-^  +  h){a- 

^-h). 

23. 

(a-^_a-i)(a-^-2a-^). 

15. 

^a-^  +  3)(a- 

^-3). 

24. 

(a-^6  +  c-i)(2a-i6-c-') 

16. 

;a-2-4)(a- 

-2-1). 

25. 

(a-i  + 6-^  +  1)2. 

17. 

:a^-2)(ai 

+  2). 

26. 

(a-2  +  6-2_c)2. 

Divide : 

27.  a-3  +  3a-2  +  3a-^  +  l  by  a-^  +  l. 

28.  a"^4-2a"^  +  l  by  a"*  4- 1. 

29.  a  +  6  by  a^  H- &^. 

30.  a—  125 c"^  by  a^  —  5c"i 

31.  a*-6a^  +  12a^-8  by  a^-2, 

32.  x^-\-a~^  by  a;^4-«~^. 

33.  a'— 3a^  +  3a"^-a"^  by  a^  —  cC^. 

34.  10x-''-27a;-3  4-34a;-2_18a;-i-8  by  ^x'"" -Qx'^ -2. 

35 .  12  a"^  - 17  a"^  -  9  -f  13  a^  -  63  aHy  4  a"^  -  3  +  7  al 


36. 


6  a~i  +  11  a  ^  V  » = -^-z  +  10  ic^  by 

Va       Va 


—  +  5  a-^a;^  -  2  a;l 


90  EXPONENTS 

Divide  the  following  by  inspection : 

37.  (a-2  -  9)  by  (a-^  +  3).  42.  (a  -  8)  by  (a^  -  2). 

38.  (ci-2-6-2)  by  (a-i-6-i).  43.  {a^  +  27)  by  {a^  +  3). 

39.  (a--*  -  16)  by  (a-^  -f  4).  44.  (a-^  -  64)  by  {a^  -  4). 

40.  (a  -  81)  by  (a^  -  9).  45.  (a"*  -  fe-*)  by  (a-^  -  ft-^). 

41.  (a-'-b-^)hj  (a-^-b-^).  46.  (a--*  -  16)  by  (a-^  -  2). 

Factor : 

47.  a-2-6-2.  57.  aj-*  -  9 a;-^  +  8. 

48.  a-^-81.  58.  x-^-Sx-'^-ASx'K 

49.  4a-2_256-^  69.  2  x'^ -{- x-^  -  10. 

50.  a^-9.  60.  4a;"^  +  lla;'^-3. 

51.  a^-8.  61.  a;^-27. 

52.  a-2  +  5a-^  +  6.  62.  a;^  +  64. 

53.  a-2— 6  (1-^-^  +  5 2/-^  63.  a;^  -  8. 

54.  a^  — lOa^  +  25.  64.  a;  — 4. 

55.  a^  +  8a^.4-16.  65.  Sa?-\-b\ 

56.  m"^-5m"^-36.  66.  a-i  +  1256-'. 

Simplify : 

67.  a-'^b-\  ^4  (a  +  l)(a-l)-i4-l 

68.  a-'  +  b-\  '  (a -\-l) (a -!)-'-! 

69.  a-'b  +  ab-\  75.  ^-(g  +  ^)"\ 

l  +  (c_l)-i 

70.  a-^bc  +  ab-^c  +  abc-\ 

71.  a-\a  +  b)  +  (a^b)b-\  76.  ____^--^_^. 

72.  (a;-l+rO-^(a5-^-rO•  ^^  a(a  _  l)-i  4.  6(a  +  l)-i 

73.  (l+mn-i)-7-(l+m-^w).  '  a(a+ 1)-^  + 6(a-l)-i* 


EXPONENTS  91 

78  mn-^  +  nr^n  ^^     x(l  -[-  a;)-^  +  x-\l  —  x) 

m~^  —  m~^n~^  +  n~^  '    x(l  +  a;)~^  —  a;"^(l  —  a;) 

80.  ra-|  +  (m  +  n)-n|-j^  _^  ^^2  ^  ^2  _  a')2-'m-'n-'l 

81.  [(m  +  a)~\m  —  c)~^  +  (m  —  a)-^(m  +  c)"^] 
-s-  [(m  +  a)"\m  +  c)~^  +  (m  —  a)-\m  —  c)-^]. 


Expand : 

82.  (x-2x-y.  87.    (V^-3V^)*. 

83.  (2a;-i  +  3)^ 

84.  (x-^-Sa^*. 


88. 


85.  (a.-f2a.-0^  ^^     ^__^       ^^. 

86.  (Va;-A/a;)3. 


Extract  the  square  root  of : 

90.  x'^ -10  x-^-\- 25. 

91.  a;-8a;*  +  18ic^-8a;i  +  l. 

92.  9a-2-6a-i  +  13-4a  +  4a2. 

93.  9a;-*-30a;-3  +  67fl;-2-70aj-i4-49. 

94.  4a;^  — 4a;^+13iB^-6a;^  +  9. 

95.  9(B-12a;^  +  34a;~^-20a;~*  +  25a;-l 

96.  16a;-^--^7^-7  +  12^/aJ  +  4^/a^. 

■y/x 

^^     9  a     24Va     24 V6  ,  9  6  ,  ^^ 

97.  -7 ;= ;^H l-o4. 

^         V6  Va         « 


92 


I 

EXPONENTS 

Solve  the  following  equations : 

98.    a;-i  =  2. 

105.    a;~^  =  -8. 

112. 

X~n  =  —  2. 

99.    ic^  =  3. 

106.    xi=^. 

113. 

--*  =  A. 

100.    x^  =  -2. 

107.    a;"^=-i.. 

114. 

»="*  =  t1t- 

101.    x-^  =  2. 

108.    x~^  =  l. 

115. 

1 

102.    a;~^  =  — 3. 

109.    a;"^  =  16. 

116. 

V5="^=iooo 

103.    a?*  =  8. 

110.    a;"  =  2. 

117. 

x^  =  </K 

104.    a;^  =  -27. 

111.    a;"  =  2*". 

118. 

xt  =  V^. 

119.    (a;  4-1)'  = 

4. 

127. 

(a^ 

-l)-2 

=  i- 

120.    (a; +  2)3  = 

125.                  128. 

(2 

0.-1)- 

•'=.v 

121.    (a;-l)^= 

=  3. 

129. 

(X- 

-^  +  1)- 

''  =  9. 

122.    (a? -5)^  = 

=  1. 

130. 

(X 

■^  +  2)- 

■^=16. 

123.    (a;  +  l)^  = 

:4. 

131. 

(a.- 

■t-5)- 

-^  =  i. 

124.    (a;-3)i= 

z8. 

132. 

(a:- 

■^-i)- 

■'=h 

125.    (a;  +  4)-3: 

=  27.                  133. 

(X- 

-^  +  3)- 

-"  =  1. 

126.    (3aj-l)- 

2_ 

:i                      134. 

(X 

■|_7y 

=  1. 

Find  the  value  of 

X  in  the  following : 

135.    4*  =  8. 

138.    9^  =  27. 

141. 

8'  =  ^T. 

136.    4*  =  64. 

139.    9^  =  ^V 

142. 

32-  =  9. 

137.    16^  =  8. 

140.   27*  =  3. 

143. 

a)-^=8. 

Find  the  value  of  x  in  the  following : 

144.  x-^  =  y',  y^  =  4:.  148. 

145.  x-^=y',  2/^  =  2.  149. 
-^  =  -8.  150. 
-">                      151. 


146.    x'^  =  y-^', 


147. 


3.-1  =2/*  J  2/~^  =  -2. 
«"^  =  y-^ ;  2/f  =  4. 
a;"^  =  y~^ ;  t/     =  '^' 


BABICALS  93 

Find  the  value  of  n  in  the  following ; 

152.  2«-i  =  16.  157.  3"+!  =  ^. 

153.  3'*-i  =  27.  158.  4'*-2  =  J[^. 

154.  9"-^  =  27.  159.  (i)'*-'  =  ^. 

155.  4"+'  =  16.  160.  (i)"-'  =  A- 

156.  16~-^  =  8.  161.  (1)"-'  =  ]^. 

RADICALS 

I.    TRANSFORMATIONS 
lizercise  47 

Reduce  to  the  simplest  form  : 

1.  V8.  14.    --v/128. 

2.  Vl2.  15.    -v^^Si. 

3.  V20.  16.    \/32. 

4.  V28.  17.    -^162. 

5.  V27.  18.    --^96. 


6.  V45. 

7.  V48.  20.  Vo^. 

8.  V72.  21.  VaFb\ 

9.  -Vl25.  22.  Va^¥?. 

10.  ^16.  23.  -Vl8^. 

11.  ^24.  24.  V27V- 

12.  -a/54.  25.  ^54^i¥. 


27. 
28. 

-■v/320a%V. 
3  V27  a. 

29. 

4V28a«6^ 

30. 

2  V56  m%3^ 

31. 

V20  m^*. 

32. 

-2-v/250a* 

33. 

^16  a^^«2/^«. 

34. 
35. 
36. 

3-V/64  m^n. 

i  V54  a^ 

37. 

-^V20  6c^c?. 

38. 

-v/54  ai«6^. 

13.    V108.  26.    -V128mV.       39.    -\Vi25<^', 


94 


RADICALS 


40.    i/27a''3^. 


41.  aV(a  +  c)^ 

42.  -aV3a\a-{-iy. 


43.  5mV(a-l)l 

44.  -a^a%a-iy. 


45.    Va«  +  2a2a;_{_aa;2^ 


46.    V36(a2-»^)(a  +  x). 


128a^ 
9c2 


48 


■V 


49.    m^ 


50. 


aV 


25  ar^ 
3^ 

108^ 
49  c^ 


63 


64. 


65. 


Change  to  entire  surds  : 

51.  2V5.  54.    2^/5. 

52.  3V7. 

53.  4V3. 

2    3/9^2 


a' 


a    /- 


3    /o- 
—  V2a. 

m 


Keduce : 

73.  Vi 

74.  V|. 

75.  Vi. 

76.  VS. 

77.  V|. 

78.  V|. 


55.  3^4. 

56.  2^7. 


66. 


67. 


3a 

2 


57.  2aVa. 

58.  8a-Va. 


60.  3a2^a2 

61.  2a-\/3aP. 


59.    -2a Va.    62.  -2aV5a. 


"2^. 


69.    -^'^9. 


2_a3/_3_ 
3\2a 


68.    -^V^l 


70. 


71 


72.  -(«-l)^5?I• 


79.  V|. 

80.  Vf. 

81.  Vf. 

82.4 
'a? 

83.    Vf- 


84.  ^. 

85.  -^J. 


86 


87.     - 


a4-2> 


a  +  &  3 


3/n~ 


88. 


89. 


90. 


2Ai^. 


BADICALS 


95 


91.    -3^^. 

4:  ax 

27  * 


92.    i^ 
93. 


95. 
96. 


_9_    /4_ac 
2a^'   3  ' 


99 


A/l2a2 


«  3  aV 

3\^- 


94.    -AJ/-^ 
3a  ^  8 

Simplify  the  indices  of : 


2c^  8  127  a* 
3a\    c2   * 

97.    iV^. 


100.  V6|(m-n)^ 

101.  -y/^ia+iy. 


4/ 9" 


103 

104.  -Vc^. 

105.  a/9. 


106.  -v^. 

107.  -^^25"^^. 

108.  ^?d^V^. 


102.    VW^^^^ 

109.  -v^QoV^. 

110.  -^/Sl  aVd". 


112.    V27aV2. 

Change  to  radicals  having 

114.  V3  and  a/5. 

115.  V5  and  -y/W. 

116.  a/9  and  -s/l, 

117.  Vl5  and  -v/SO. 

122.    Vm, 

Which  is  the  greater : 

123.  2V3  or  3V2? 

124.  Vil  or  -\/30? 

125.  2V3  or  a/42? 

Which  is  the  greatest : 
129.    V5,  a/10,  or  a/IS? 


111.   Vl6a*b'c\ 
113.    a/100. 


the  same  index : 

118.  Vn  and  A^SO. 

119.  a/25  and  a/75. 

120.  a/6,  a/15,  and  a/35. 

121.  A^,  a/7,  and  a/10, 
a/w?,  and  a/wi^. 

126.  3a/5  or  SVS? 

127.  2a/4  or  a/10? 

128.  a/|  or  a/|? 

130.    V6,   \/16,  or  a/35? 


96 


RADICALS 


Collect : 

131.  V50H-V18-V8-V32. 

132.  Vl8~V98+V50-V72. 

133.  V27-VI2+V75+V3. 

134.  Vl2a-V27a-V48a  +  Vl08a'. 

135.  Vow^  —  Va%i  +  VOom^  +  V4 a^m. 

136.  |Vl2-V50  +  ^V48-Vi8. 

137.  V20-V|  +  V|  +  4V2-3V5. 

138.  Vl24-Vi-V27  4-V|-Vl08. 

139.  V50-^-6Vi  +  3V|. 

140.  -^i-Vi  +  V98-2V27. 

141.  2-v/|  4- 3-5^ -2^/144. 

142.  2V|-3VS-V}4-Vi000. 


143.    3Vo^  +  4 


25 


4/aV 
81 


144.  V50-v''432+V32  +  ^250. 

145.  3^4-^24-^3+A/i6. 

146.  30VJ-fV8  +  9V84l. 

147.  VT80-2V5  +  15V|. 

148.  |Vl62  +  10V4|-13V2. 

149.  -t/36-V|-4V6  +  2VJ. 

150.  10Vl2}  +  7V2-3V338  4-5^  +  4Vf. 

151.  V24-6Vi  +  iV96-V66|  +  |V¥- 


RADICALS  97 

152.  6V33j-V96  +  V|-|V-^  +  4V6. 

153.  12Vi6j  +  5V3-5V432  +  6V^. 

Multiply : 

154.  (V20-fV80+V45)  by  V5. 

155.  (V8-2VI2  +  V2O)  by  V6. 

156.  (^-1^32  + -J/5)  by  ^/16. 

157.  (2V3-2)(2V3  +  2). 

158.  (2V5  +  3V2)(3V5-4V2). 

159.  (5V3-2V2)(3V3  +  4V2). 

160.  (V2+V3)2.  163.    (3V2-5V3)2. 

161.  (V3-2V2)2.  164.    (V3-V2)^ 

162.  (2V3-2V2)2.  165.    (2V2-2)^ 

166.  (3V2-2V3)3. 

167.  (V7-V2  4-V5)(V7+V2-V5). 

168.  (3V|  +  3Vi-10Vi)(iV24  +  iV75  +  V20). 

169.  (  VlO  4-  Vl9)  ( VlO  -  Vl9). 

170.  (Vl3  -  2 V22)  (Vl3  4-  2V22). 

171.  ( V2  -\-x-{-Vx)  (V2Tx). 

172.  ( Va  +  1  -  2)2. 

173.  (2  Va^"^^  -  3)2. 

174.  ( Vm  +  1  -  Vm  -  1)  ( Vm^^). 

175.  ( Vm  +  2  -f-  Vm)  ( VmT2  -  2 Vm). 

R.  &  S.  EX.   IN  ALG.  — 7 


98  RADICALS 

176.  (m2  +  mV3  +  3)(mV3-3). 

177.  (Vm  —  Vm  —  n  +  Vn)  (Vm  +  Vm  —  n  +  Vw). 

178.  (Va-l+Va  +  1)-. 

179.  ( V2I  -  6  V3)  ( V2I  +  6  V3). 

180.  (5V?T^-4V?^^2. 

181.  (2V^  +  V4^^)(2v^-V4"=^). 

182.  (Vo+I - V2)(2VaTl  +  V2)(2a+ V2a  +  2). 

183.  VS--\^-W-  185.    Vi|--v/W- 

184.  VS--\^||-  186.    (V2-2^/4)(2V2-^/4). 

Divide : 

187.  2V32  by  3Vi20.  193.  \V^  by  ^V^. 

188.  \/8l  by  V3.  194.  (5V18-3V27)  by  3V5. 

189.  ^J/12  by  4V2.  195.  (2V54+iV24)  by  3Vi. 

190.  VU  by  ^32.  196.    Kk  by  Ajji. 

5  Vc      "^  10\a 

191.  -Wc^  by  -ySom^.  r- 

192.  (^12  +  4VI8)  by  6V2.    '''•    (IV^-^V^  ^^  ^• 

198.  (12V5  -  8V15  +  3 V30)  by  6VIO. 

199.  10V3  -  15V42  -  9  V2  by  5 V6. 

200.  (5^^^  4- 3^/45 +  6^/30)  by  2\/i8. 

201.  (</32--v^'48-a/80)  by  </3 

202.  Xi^,  by  f-^^j:K:Y 


RADICALS  99 

n.  MISCELLANEOUS  APPLICATIONS  OF   RADICALS 
Exercise  48 

Extract  the  square  root  of : 

1.  6  +  4V2.  7.  44-16V7.         13.  16  +  2V39. 

2.  11  +  6V2.  8.  30-12 V6.  14.  74-6V77. 

3.  28-10V3.        9.  88-16VI0.        15.  77-28V7. 

4.  21 +  8 Vs.         10.  57-f-12Vl5.        16.  o?+h  +  2a^h. 

5.  45-20V5.      11.  207-40Vli.       17.  o? +  2c^-2a^/2~c. 

6.  42  +  12  V6.        12.  82  +  12 V42.        18.  m"^ -\- m  +  2 m^Vm. 
19.   2a;  +  2V^^^.                 20.   m^H- (2m-2)V2m-l. 

Rationalize  the  denominators  of : 
,1.    A..  29.    -1-.  37.    2Va  +  3Vc- 


V2  V3-2  2Va-3Vc 

22.  A.  30.    -^ 38.    5V2^-V6j. 

V3  V2+V3  5V2a;  +  V6a; 

23.  A,.  31.    2  +  V3  g^     4V2^-3V^, 
V2  *    2-V3  *   3V2^  +  5V» 

24.  11.  32.    ?V|±2  ^^     aVc-ftV^, 
'    V6  *   2V2-1  *   aVc  +  6Va; 

V24  ^,     V5-V3  a  +  V^^^=^ 


25.    -^^-^^-  33. 


41. 


V3  V54-V3                 *   a-^d'-l 

26.  ?^.  34.    ^^  +  ^^  42.    V^-3+V^. 
Vi2  *    Va-V6                 *    VaJ-3-V^ 

27.  -^.  36.    2V3  +  3V2.  ^3 
2+V2  2V3-3V2 

28.  ^.  36.    3^-2^^.  44. 


V3^ 

;  +  l+V2a- 

-1 

V3^ 

'  +  1- 

-V2a- 

-1 

Vm^ 

-2- 

-Vm2  +  2 

3-V3  2V5-2Ve  Vm2-2  +  Vm=^-f-2 


100  RADICALS 

Find,  to  three  decimals,  the  numerical  value  of: 
45.    ~  47.    -^.  49. 


V3  2V3  2^2 

46.    A.  48.    A.  50.    ^^^±1. 

V5  -v/2  V2-1 

53.  ?V|±3^|.  ^^^ 


51. 

V3  +  1 
V3-1 

^9. 

V3  4-V2 

V3-V2 

5V3  4- 

■2a/2 

2v3-2V2  V2-V3 


Simplify : 


55.    Vl5-6V6. 

V8 


(V6-V2)(V3  +  1) 


58     Va;  +  2  +  Va^-2 


59. 


Va;  +  2-V«-2 

1  .  1 


a  —  Va^  —  4     a  4-  V a^  —  4 


60.  ^    _     +  1 


(3-V2)2      (3+V2y 


g^        2V^-i       ^       3V^"=i: 


3Va;  +  l-2Va;-l     ^^x  +  l+2Vx^^ 


^^     V26  4-8V3_ 
V6- V2 


63.   :^^19-8V3, 
V3  +  4 


KADICALS 


101 


64. 


65. 


66. 


67. 


68 


Va  4-  V^     Va  —  Vb 


2-Vb 


2Va 


V3-I-V2-V2 
V3  +  V2  +  V2 
a;  +  5  ic  — 5 


( Vll  +  6V2)  +  (^6  +  4  V2)^ 
5  -  2  V2 


\  ^  a*     a      ^  a;     ay 
70.    [3V3+(V28-16V3)]2. 


71. 


72. 


73. 


74. 


(a;  +  V4-^(a;-V4-arO 


V34-2V2-V3-2V2 
V3  +  2V2+V3-2V2 

3-V2 


Vo;  4- 1  +  Va.'  —  1 
Va;  +  1      1 


Va;-1 
34-V2 


V3+V3+V2      V3-V3+V2 

a  +  Va   r  a-4       (a4-2)Va"| 
«  («  +  2)  LVa  -  2        a  H- Va   J 
[Hint:  a  -  4  =( Va  +  2)(V^  -  2).] 


75. 


102  RALICALS 

76.  Show  that  — - —  =  2  +  V2. 

2-V2 

77.  Show  that  ^  +  ^^~^^=V5-2V6. 

2_V2  +  V6 

78.  Show  that  ( V24  + 16 V2  -  Vll  +  6^/2)^  =  5.82842  +. 

79.  If  a;  =  4  +  3  V5  and  2/  =  5  +  2  V5,  find  value  of  (a^  _  /)2. 

Solve: 

80.  V^+T  =  Vl2.  85.  2(V^-3)(V^  +  3)=3. 

81.  V«T3=V2a;-10.  86.  Vx^^ - V^^^  =  V2. 

82.  3Va;-9  =  2Va;  +  ll.  87.  Va;  + 3  +  Vx- 2  =  5. 

83.  2V^T3  =  3VlO-a;.  88.  V3H-V9i»  + 1  =  3V^. 

84.  VflJ  4- 13  =  13  —  V^.  89.    Va;  +  9  mw  =  3  m  +  V^. 

90.  </10a;-6=V2. 

91.  Va;  +  3+V9aj-l=V4a;-l. 

92.  V2a;  +  l-V8a;-l=V2a;-l. 

93.    ^j7  +  ^^S  +  ■^^  =  3.  95.    ViC  +  7+V^  =  ^. 

94     V^  +  3_V^-f-9  __^__ 

V^-2      V^  +  l"  96.   V2+V^^^  =  V^^^. 

97.  —  Va;  —  3  =  V^. 
Va;-3 

98.  V^  +  V^"^ -5__  =  0. 

99.  2a^  = ^ 4-V4a;-3. 

V4a;-3 


IM AGIN  ABIES  103 


100.    V^^^  +  V^  =  _6. 

Vx  — 5  — Vic  • 

VaJ  +  m  —  V^  _  1 
Va;  +  m  +  Va;     ^* 


102. 


103.  V2a;-a2-f-V2a;  +  a2  =  26. 

104.  Va;-2a=    ^^       -  V^. 

Va;  — 2  a 


105.  V2^  — Vm  =  V2a;  — Vm(5m  +  8a;). 

106.  J^- J^  =  0. 


107. 


a;  +  4      ^a;  — 4 
3VS  +  2     3V^+1 


2V^'-7     2V^-5 
108.   2-\/x  -  2  -  3 Va;  +  2  =  5 V^. 


109 


Va  ,        «  /- 


Simplify : 


IMAGINARIES 
Exercise  49 


1.    V^=^.  5.    2V-81.  9.    -2V-36a«. 


2.    V^16.  6.    V=^.  ^^-    V-(aj  +  2/)l 

4.    V^lii.         8.    -V^9^^.     ''•    -^^-("  +  ")^- 


104 


IM AGIN  ARIES 


Collect: 

13.  V 

14.  V 


4  +  V-94-V' 
"9  4-V^^^36 


25. 


V-49+a/^64. 
15.    V^^36  +  V^=32r-V^^100-V^169. 


16.  2V-9 

17.  3V^=^-V 


3  V^^16  +  V-49  -  V- 25. 
V^^144+V^256. 


81 


18.    iV 


4  H-  2  V-  36  -  \-\/-  25  +  iV^^ST 
19.   2V^  +  3V^ 

v 


2V- 


^  +  3V-^V 


20 


+  2V-4a2-V-9a2  +  V-16a2. 


21.    V-4m2-iV-16m2-h2V_i21m'^-V^ 


m'' 


22.    aV 


Multiply : 


2 


a 


16  a^ 


23.  V^^  by  V^T. 

24.  V^=^  by  V^^. 

25.  V^^  by  V3. 

26.  2V^^  by  V^^. 

27.  3V^^  by  V^^. 

28.  V^^  by  2^^^. 

29.  3V^^  by  3V2. 

30.  _  2 V^=^  by  V5. 

31.  2V^^  by  -V6. 


32.  3V^=^  by  2V^^. 

33.  3V-27  by  -V^^. 

34.  -  2  V5  by  3  V^=^. 

35.  a^—a  by  Va. 

36.  —  aV— a  by  —  V—  a. 


37.    -2V-3a  by  3V^^. 


38.   — 3aV— a  by  5V— a^ 

39 

40 


—  2aV— 2(1  by  aV277. 
-3aV"=^by  -2aV^=^. 


41.  V^^  by  V^6  by  2V^=^. 

42.  V^=^  -  V^^n^  +  2  V^^15  by  2 V^^. 

43.  (3+V^  by  (3-V^^). 


IMAGINAEIE8 


105 


44.  (o-V^(5+V^. 

45.  (2-V3)(2-V^. 

46.  (2+V^(2  +  2V^^). 

47.  (5-2V^^)(3-3V^=n[). 

48.  (3V2-V"^^)(2V2-3V^2). 

49.  (4-3V^=^)(2+V^^). 

50.  (Vir3_v^(V^^+V^^). 

51.  (2V^5-3V^(3V^=^  +  2V=^). 

52.  (3V^=^  +  2V2)(4V^:^-3V2). 

53.    (3-V^)^        54.   (2-3 V^'.        55.   (2V3-2V^' 

56.  (a-l-V^l)(a-l+V^. 

57.  (a  +  &\/^l)(a-6\A=3). 

58.  (2  a  +  2  6  V^^)  (2  a  -  2  h^^l). 

59.  (V^l+V^2-|-V^^)(V^=i:4-V^=^-V=^). 

60.  (V^^-V^=^  +  V=T)(V^=^+V^^-V^. 


61. 


62. 


63. 


Rationalize  the  denominators  of 
3 


12 

V^=:3' 


64. 


65. 


66. 


V-15 

2V3 

3V^=^ 
2V^ 


67. 

3V-9 

-6V-3 

68. 

Vio 

V-2 

RClk 

Vl5 

2V-3 


70. 


71. 


72. 


-5V12 


Va 


73. 


1-V=2 


74. 


34-V-2 


75. 


V2-Vi:3, 
V2+V"^ 


106 


IMAGINABIES 


76. 


77. 


78. 


V3  4-V^2 

1-V^  * 

vCis  +  v^ 


79. 


80. 


g  +  V-l. 
a-V^l* 
m  +  nV— 1 


82. 


83. 


m  —  n- 


2a-|-6V^ 

3a-26V^^ 
2a-6V^ri 


2-3V2     e,     V2a+2V-2a    ^^     a-vT=^ 

ol.     ^ — — — — •    o4i 


2V-2H-3V2 


Va— V— a 


a  — Va  — 1 


Simplify : 

85.  (V^l)*.  87.    (V^*.  89.    (V^)-*. 

86.  (-V"=3)^  88.    (-V^=^/.  90.    (-V^^)-^ 

91.  (1_V^'-(1-V^^)^ 

92.  (3-V^'-4(5-V^^)'. 

93.  (l+V^'-Cl-V^'^. 

94.  (1-V^^ 

95.  (1_V^^-(1-V^^+(1-V=^). 

96.  2(V^n:)3-2V^(V^=3-l)^ 


97.    = (l--\/^^y.  99 

V2-1  ^ 

98.  (1  +  V^)--(1-V^1)-.  100. 

2V^1 


iC+V  — 1  iC— V— 1 

m  +  ^V— 1  ,  m  — nV— 1 


Find  the  square  root  of : 

101.  2-4V^^. 

102.  1-56V^^. 

Kesolve  into  imaginary  factors : 

105.  a  +  b.  107.    a +  2 6. 

106.  a +  4.  108.    a2  +  4. 


m  --  71 V—  1     wi  +  n  V— T 

103.  32-32V^. 

104.  -3-12V^^. 


109.  a^  +  1. 

110.  2«2  4.3. 


GENERAL  REVIEW 


Iizerciae  50 


1. 


3. 


Find  H.  C.  F.  and  L.  C.  M.  of : 
J  m^  +  6  m"  4-  5  m  —  12, 
[m^-3m^-22m-12.        5. 
6m3-llm2-14mH-24, 
8m»  +  18m2-llm-30. 
lOa^  +  a^-hlSa'-S, 
16a^-a'-\-2. 


l-8a;-3, 

l  +  9x-^-22ar*, 


4. 


rioa^ 

l8a^H-4a*^-2a2»_l. 


1 3  a-1  -  4a~*  - 13  +  14a^, 


Simplify : 

2  mn(m -^  n)~^  —  m  .  2  mn(m -\- n)~"  —  n 
n-^ -\- (m  —  2  ny^        m^^ -{- (n  —  2  m)-^ 


i-1 


9. 


(g  - 1)  [3  g  +  (<^  -  l)n-^-(l-3  a-{.a^(a^^l)-^-(a-l)-\ 


(I_2g  +  g2-2a«)(l  +  2g  +  :^g2  +  a3)-i 

10.  [(m  — n)(m  +  w)-^  — (m  +  7i)(m  — n)-^] 

11.  1+ K, 


-W-2). 


2- 


1  + 


1 


2  + 


3-2a? 
a;-l 


12.    If  -  =  ±,  show  that 

n      y  n  y 

107 


1^         'T- 

GENEBAL  REVIEW 
1                        1 

^-^^  +  1^-^ 

14.  (a-l)(a  +  3)-i-l-(a-2)(a-3)-i 

+  [5(3-2a)-a2][9^a2]-^ 

15.  m^-f 


7M' 


m2  + 


^/i"" 


??i^  +  m^ 


mr 


16.  [a-6(c-a-i)-i]-\ 

17.  If  «  =  ^,  show  that    '^^  +  ^  =  _3m±^. 

6      n  6a-f-36     5m  +  3n 

18.  Show  that 

a^  — 2/^        '     (c  — 1 


Solve : 


1  + 


19. 


3ic4-l 


x-\-^ 


x-1 

ir  +  4 


1  + 


xy 


ix-yf 


2a^  +  a;-3     x-2x'  +  Q     2-^x^o? 


20     3a;-l        a;  +  l     ,  ^a;  +  4^4a;  +  5 
8  9a;-16  4  6 

21.    ^±l-f.(a;-l)(a;-2)-i     ^^~^ 


cc  +  2 


x-1 


6 


23.    5a; 


=  .-[3-{ 


aJ  + 


a;-3 

15   * 
2 


(7  a; -12)-^  [_'       [      '  (3-a;)-M_ 

24.  [l+a;(l-a;)-i][l-a;(l4-a;)-'][l-a;'+(l-a;'K^]=3. 

25.  Show  that  -^  satisfies  «-4a;_a  +  2x^^ 

5  2a  — a;     x  +  2a 


GENERAL  REVIEW 


109 


26.    Solve  (m  +  x){n-\-x)~m{n-{-p)  =  '^^  +  oi:^. 


X      ex 


27.    Show  that  cd  +  -,  =  —  +  «  when  x  =  ad. 
d      a 


Solve  the  following 


28. 


29. 


30.  < 


31. 


=  3, 


32. 


x-5     2x-y-l_2y-2 
4  3  5    ' 

2y-^x  —  l_x-\-y 
9  4    * 

x  —  2     x-\-  y—1  _ 
3  4 

X -\-S     x  —  2y—l_^ 
4  2 

2a;  +  l  2-Sy^l 

3  5          6' 
3y-2  2a;  +  3_3 

4  6  8* 

( 4:X  —  Sy±l  __x-\-Sy 
I  2^  T"' 

3a;  +  2y      j^^2a;-3y 

5  3 

I  ^x-V±y  _  2        x  +  1. 
3  ^         2    ' 

14  ^       ^^      2 


36-  -I 


37. 


38. 


39. 


f^     i_^_13 
2a;"^32^  3* 

13a;     2y     27* 
?/     a;     3a 

1+1=A. 

x     y     2a 

Sx-{-4:y  —  5z  =  2, 
2x-Sy-Sz  =  -9, 
Sx-y-2z  =  0. 


(2x-y-\-Sz  =  -i, 
40.  -!  Sx-3y-2z  =  ^, 

\4.x  +  2y-5z  =  5i. 

(  cy  +  bz  =  2, 
^^    (ax-by  =  a^-b^-2ab,      ^^    I  ^,_^,^^2, 


33.   2aj  +  32/  =  3a;  — 22/  =  l. 


35.  i 


bx  -\-  ay  =  2  ab  -\-  a^  —  W. 
a-^b     a  —  b 
4a6 


bx  +  ay  =  2. 
[       x-'-y-'-z-'  =  S, 

I     y-^-x- 


110  GENERAL  REVIEW 

Expand : 

43.    {2x-ay.  44.    [3a  -  {2a+ (5a-2^"=l)i]^ 

45.  (      "^-^^      Y.  47     ^..--^^Y 

46.  {a-^  +  2x-y.  48.    ("v^ ^-Y- 

Find  the  square  root  of: 

49.  a;-'^  +  4a;-3'»-2.'B-2'»-12a;-~-f-9. 

50.  a^-2a2»'-ll  +  12a-2-  +  36a-^. 

51.  4-4a^-lla  +  14a^  +  5a2_i2a^^.4aS. 

52.  (a;-2  -  4) {x-^  -  3  a;-*  +  2) (a;-^  +  a;"^  -  2). 

53.  (m  +  m-y-4(m-m-i). 

Find  the  numerical  value  of: 

54.  V.073  to  5  decimal  places. 


55.  V.0073  to  5  decimal  places. 

56.  Vs  -  3  VXiM  to  3  decimal  pla<;es. 

57.  V.007  +  .3  VlAi  to  3  decimal  places. 

Simplify : 


58.  ^!i^\^/:r2iw. 

59.    [-3-v/-27a-2-^(- 8-')-i]-'. 


64.  f-s/-S(^d- 

65.  64-^-flV'</Sr^^ 


GENERAL  REVIEW  111 

67.   (SaV^-h27 x^Vcir^yi 


\  16  ac'VdJ 


68.    (a?"*)"-^ .  (a^)'»+l(a;'")l-2^ 

r 


n+l 


_        _  69.    Jinh^.x^Va'n 

66.    (m^)  «  (m^)  ^   -- (m-y\  \^  '^^   \ A^-v/n' 

70.  V25  a-ift V^(27-ia"V^^aa;^6-i)i 

71.  [7a;-V^]«  •  [3a^2/"*2;]-^  "^(s^^^T* 


Collect: 


"■   (1)^3^ 


73.  7-1  +  C^y  -  (7  a;)«  -  49"^  -  2-2. 

74.  81-^-5a;«  +  9(3)-3+(125-V^  +  rt 

75.  (^:^J_V3T27^  +  (-243)i  +  (:^J- 


Multiply : 


76.    i-l  +  ft'^by  a-2  +  l+r^- 


77.  a3-3  +  3a-3-a-«  by  a'---^--- 

a     a* 

78.  J^_4aj-i  +  ^-24by  ^+4^  +  5. 

Divide : 

79.  a-^—b  by  a"*— &i 

80.  27  m-3  -  8  w2  by  3  m-i  -  2  nf 

81.  a;^+2a;^-16a;"^--32a;-^  by  a;*+4a;'^+4a;"^. 

82.  a--^  +  a-2'"62n  _|_  54n    by       1    _^_j_?,2n 


112  GENERAL  REVIEW 

Simplify:    83.    (oT +  2  a'^^y  -  {a'^  —  2  a-'^f. 

84.  [(a  +  6)^  +  (a-6)^]2. 

85.  [(7/i-l)*+(m  +  l)^J. 

3x  _35 

86.  (m'^-m  ^)-^(m' -^l-[.m-''). 

Collect:       87.   f VlGj  +  ^ - yV V432  +  ^ VIp. 

88.  5V75--jVi47--^  +  2VJ-^81. 

89.  Multiply  ^  by  -^^  by  V|. 

Simplify:    90.    (3V6-2V3 +  5V2)2. 

91.  (2V5-3V3-2V2)(2V3+V5-3V2). 

92.  (^9-6V|  +  V48)2. 

94.  (V3-V2)2-2(2-V6)(2-V2). 

95.  (3V2-V3)(2V2  +  V3)(3V3-V2). 

96.  (?^-5^  +  iO.V^25-^216). 
VVlO       V6       V2/  ^ 

97.  V52-6V35. 

98.  V2a  +  l-2Va2  +  a. 

99.    (^/2-V5)^.  102.    (V2  +  ^3--^)2. 

100.  (V2-^y.  3^-4^81 

101.  (2V2- 3^2)2.  ^^^-         -^243 


GENERAL   REVIEW  113 


Kationalize  the  denominators  of  the  following : 
104.    -^ -•  106. 


3V6-2V3  '    V3-V2  +  1 

105.     — ^- -.     •        107.    '^  +  "^- 

Find  the  numerical  value  of: 


^  V5  +  V2  V2 

110.  Which  is  the  greater,  V5  or  -^/\l  9 

111.  Which  is  the  greatest,  V|>  \/f,  or  \/|? 

112.  Show  that  V|  >  ^f . 

Simplify : 

115.  If  m  =  i  ( Vc  +  d  +  Vc  -  2  d)  and 

yi  =  ^  (  Vc+c?  -  Vc-2d),  find  the  value  of  m^  +  n\ 

116.  Simplify  by  inspection : 

(Vm  4-  71  H-  Vm  —  Vn)  (Vm  +  ti  —  Vm  —  V^). 

117.  Change^— J^i-II-^ to  an  entire  surd. 

c  —  cZ  ^  c  -f  fZ 

118.  If  a  =  11,  6  =  —  12,  and  c  =  3,  what  is  the  numerical 

value  of  ,         /-To A — 

—  0— Vo^  —  4ac^ 

2a 

119.  If   71  =  11,    a"=5V2-2V3,    d  =  -(V2+V3),   find 
value  of  ^[2a  +  (7i-l)d]. 

R.  &  S.  EX.   IN  ALG.  — 8 


114  GENERAL   REVIEW 

Simplify : 


■«         X  —  1 

x  +  1 
Vx  —  Vy      -y^  +  V^ 


122. 


123,    f^^^  —  h  —  Va  +  &      Va  —  6  -f  Vg  -|-  5 


a +  6/ 


\  Va  —  6  +  Va  +  6     Va  —  b  —  V 

Solve  the  following  equations : 

124.  V»  +  V^^^  =  V5.  '^'  »' 

125.  V3a;-2  4-V3^-2  =  0.   <)  ^  '^t^ 

126.  Vic  +  6  +  Va;-4  =  2.    ~U  »  £> 

127.  Va;  +  2  +  V4«  +  l  =  V9a;  +  7.     't -^  "^ 

128.  VS^+T -  V2x  +  3 - V2a;-2  =  0. 


129.  V2  +  V4^+5  =  V2¥T3. 

130.  -^+        1  2 


V^^=^      Vx+^     Va^-4 


4 


131.  -Va;-V4+^  =  0. 
V4  +  a; 

132.    ^  4-  ■'^ 


aj  +  Va^-3     a;-V^233     3 


133. 


134. 


GENERAL   REVIEW 
2-x 


X 


V2  +  Va;      V2-Vx 
2  m  — n         V2m^  +  n 


V2 


mx  —  n 


x  +  n 


Simplify 


135.  (V^ri)3+(V^/. 

136.  (V^=^)'-(V^^ 

137.  (l_V^2-h(l+V^^)2. 

138.  (i-2V^^y  +  2(2-^^iy. 

139.  (V3  +  V^)(V3-2V^). 

140..^(2y3-3V^^ 

141.  (-|+iV33)3. 


142.    V4 V6  - 11. 

Rationalize  the  denominators  of : 
V^  _.       2V18 


115 


143. 


144. 


V^18 
V2 


145. 


146. 


-V28 


147. 


148. 


V5- 


32+2V^ 


What  are  the  conjugate  imaginary  factors  of : 

149.    m-{-2n?  150.    a^  +  T?  151.    3^2  +  2? 


116  QUADRATIC  EQUATIONS 

QUADRATIC    EQUATIONS 
I.    NtBIERICAL  QUADRATICS 

Exercise  51 

Solve : 

1.  2x^-7x  =  W,  11.  x'-Q^O. 

2.  2x'2  +  a;  =  15.  12.  x'-4.x  =  0. 

3.  3a:2_^7^.^20.  13.  a^  +  l  =  0. 

4.  6x^-19x  =  S6.  14.  3a^  =  7. 

5.  5x2  +  14aj  =  3.  15.  5ar'=llx. 

6.  a;2  +  3a;4-l  =  0.  16.  2a;-  +  3  =  0. 

7.  .T2  +  3a;4-3  =  0.  17.  5ar-3x  +  l  =  0. 

8.  aj2_5a;_i  =  o.  18.  3«2_^5^,_^3  ^q 

9.  .T2-5ic  +  7=0.  19.  5a^  =  2a;  +  l. 
10.  3a;2  4-2iK  +  l  =  0.  20.  7ar  =  6x-l. 

21.  3ar^  +  .^'-5  =  a;2-ll  +  8a;. 

22.  (2x-^S)(x-5)  =  (x-5)(x  +  S). 

23.  (3aj-7)(2a;  +  l)  =  (5a;  +  2)(2a;-3). 

24.  (2a;  -  1)  (3a;  +  5)  -  (a;  +  5)  (3ic  -  2)  =  5  -  («  -  2)1 

25.  (3x+  1)  (a;  -  5)  -  (2a;  -  1)  (3a;  +  2)  =  (a;  +  6)^  -  1. 

26.  (x  -5y-(2x-  3)2  _  (a;  +  4)2  =  a;  (a;  -  5). 

27.  (2 a;  +  l) (a;- 5) +2(a;- 3)2- a;(a; -4)  =  2(a;-|)2-15|-. 

28.  (a;  -  5)2  -  (3  -  2x)2  -  (2a;  -  1)  (a;  +  4)  +  5a;  =  0. 

29.  2(a;  +  2)  (3a;  -  1)  -  3  (a;  +  1)  (4  -  a;)  =  x(a;-  2)  -  17. 


QUADRATIC  EQUATIONS  117 

30,  ^  +  -^  =  1.  40.^-1  =  ^. 

x  —  1      x-^1  x  +  2  a;  +  4 

31     ^-1      1^      1  41     2a;-3     a;-!^     .^ 
*a;4-la;         6*  *3a;  — 2         x 

32.    ^-4  =  _^.  42.    1-1=1       -  +  2 


3-a;     5      9-2ic  3     x            2a;-l 

33.  ^^±^-^:z2^ii  43.    _J ^  +  _5_^0. 

a;_2     a;4-2        ^  a;-l      cc  +  l      2-a; 

34.  2^^  +  2^+5  =  2.  44.    -5 ^+-A_=0. 

2a;  +  l       a;  +  8  2a;-7     a;+4     a;+7 

35.  4^Zli  +  §-l  =  0.  45.    _2_=  J0_            _^. 

2a;  +  l      2      a;  x-2      x-\-2            x'-4. 

3g     _^ fl^-1^2  46     ^-3     24-a;^(a?+iy+4 

x  —  S      x-\-3        '  '    1—x    1+x         1—x^ 

2^     a;  +  3      3x-2^^  ^^     2x-l      x-2^x-S 

*«  +  5       aj  — 5  '     x  —  2       aj  — 3     a;  — 4 

^„     2a;-l      a;-2      q  ^o     2«-1     a;-7     .     3a;-l 

38. T  =  o.  48. -=4 — . 

X           x-\-l  a^H-l      a;— 1             x-\-2 

39.    ^Zll  +  ^Z:5  =  3.  49.    ^±l-^^±?+^!=5  =  0. 

a;-2     a;-4  a;-3      .t+3  ^9-a^ 

50     2a;4-l      -^       x  —  4.  __   —7x 


2x-8  2a;  +  3     9-4ar' 

2     a;  — 3     a.'  +  13_     13 

a;  — 5      a;  — 3  ~~2a;  +  5 

52  5  a;-1^2(a;4-3)      ^ 
2a;-l      a;  +  l       2a;  +  3 

53  a;  +  l  ^2a;-3      ^  36 


3a;  +  2     3a;-2  4-9a^ 


QTIADBATIC  EQUATIONS 

60.  2V^=Va;-3-f 3. 

61.  V2x-l  =  Vx-\-l. 


118 

54.  3VaJ  +  2  =  2a;-5. 

55.  a;  + 5  =  2V5  oj  +  l. 

56.  V3  a;  +  7  —  a;  =  3. 

57.  2V3 a;-f-4  =  VSa^-S a;-4.   63.    2Va;  +  l  -  V2a; -f  3  =  1. 

58.  2V^  =  a;-3.  64.   2 V3 a;  +  7  +  1  =  3 Va?  +  3. 

65.   5Va;-l-3V3a;-2=-l. 


62.    Va;4-l-l=V2a;-5. 


59.    2 Va?  —  1  =  a;  —  4. 


66.  2V3a;-2-3V^+3  +  l  =  0. 

67.  Va;  +  l+V3a;-f4  =  V5a;  +  6. 

68.  Va;  +  5-V2a;-7=  V5. 

69.  2  Va;  +  1  +  Va;  -  2  =  V7  a;  +  4. 

70.  V4a;-3-V2a;  +  2=Va;-6. 

71.  ■y/Sx-5-{-Vx^^  =  2Vx'^^. 

72.  V2-3iB-V7  +  a;=V5  +  4a;. 

73.  V3a;  +  2-V2aj  +  l=V^Tl. 

74.  V2aj  +  3-V8a;  +  5  =  -V4a;-l. 

75.  2V3a;  +  2-V6a;-3  =  3V3a;-l. 
4 


76 


.    V3a;  +  Va;  — 2  = 


77.    V3a;  +  1-V2a;  = 


^/x-2 
5 


78.    V3a;  +  3-Vaj-l  = 


V3a;  +  1 
2 


79. 


3V^qp4_V2a;-9  = 


8 

V2¥^^' 


QUADRATIC  EQUATIONS  119 


3 


80.    V3a;-5+     ^_  =  2V^^=^. 
■Vx  —  1 


81.    Va;4-2  = 


82. 


83. 


84. 


+  Var^  +  7    'aj-V^+T 


II.    LITERAL  QUADRATICS 

Exercise  52 

Solve: 

1.  2a^-5ax  =  Sa^  10.   aar' 4-aa;  +  2  =2a-a?. 

2.  6aV-7aa;  =  20.  11.   2a^  +  a2  =  a;  +  3aa;  +  l. 

3.  Sx^-abx-2a^b'^  =  0.  12.   ar'+aaj-2a;+l=2a2+a. 
44.    5 aV -24 62  =  26 a6a;.  13.    2d'x'-a^x-9ax=a'-9. 

5.  18  6V  =  3  6ca;  + 10  c^.  14.   6x'+ax+Sx=a'-{-a-2. 

6.  (B2-2aa;  +  a'  =  4.  15.   a V - a^  +  2 6a;  =  ft^. 

7.  o^-a^  +  6x  +  9  =  0.  16.    6 V+a6a;-4 ar'=2 a(3 a;+a). 

8.  4a^  =  4aa;  — a^  +  l.  17.   a^a.'^— a;— aa:^— aa;=(a+l)^. 

9.  aa^  =  3a;  +  4a-6.  18.   4:af-4.ax  =  b-a^  +  c. 

19.   4:a'x(x  +  l)  +  (a-l)(a  +  l)  =  0. 

20.  a2a^-62  =  aa;2  4-6aj.  22.   4a^-a2  =  2a  +  l. 

21.  a^a^ -  (6 -1)=^  =  0.  23.    9aV-c2  +  6c  =  9. 


il^O  QUADRATIC  EQUATIONS 

2^.    2aa^—bx  =  cx.  ^^        1       ,      1         a  +  b 

6o. \- = — 

25.  3ax'  +  4.bx  +  5c=^0.  «  +  ^     ^  +  ^        ^^ 

26.  aay^-\-2bx-\-3c  =  0.  ^q    __j^_  =  l_i^l. 

„  '   x  —  a-\-b     X     a     b 

27.  arH-pa;  +  g'  =  0. 

28.  lx^-mx-lm  =  0.  37.    ?iL±_?  +  ^Lul^  =  2|. 

2a-a;     a4-2a;        ^ 


29.  a;2-2aaj-2a;+a2H-l=0. 

30.  (a2-6>2^a2(2.'c-l). 

31.  (l-a'){x-\-a)=2a(l-x') 


30.    (a2-6>2_a2(2.'c-l).        38.    ^L_J_  1^_  2a  =  0. 


a 


39. 


a;         a;—  1 

1  111 


32.   -  —  bx  =  —  a.  a  —  x     b  —  x     a      b 

x  —  1 

o^  ^  .^     x  —  a     2a  ,  2x-\-3a 

a;  g  — 1_     a;H-2  .  18a -.  _  x-{-a 

a  +  2         X     ~x(a-\-2)         '    5a—x—Sb        ~ x-^2b' 

42.    ax^-^^^^±^  +  bx-^  =  -^^-bx. 
a  —  b  a—b 

43.   ^     1^       3a^-2a  ^^     _1-  +  -J l-=0. 

a;      (a  —  1)  (2  a  —  1)  a;  +  a     x  +  b     x -{-  c 


45.  V2ar*  — ax  — a  +  2  =  a  — 2. 

46.  V3  a^  —  4  «.«  +  1  =  2  (a;  —  a). 

47.  Va;  —  a+V3a;  —  2a=V2a;  +  5a. 


48.    V^  +  "^ a  —  Vaa;  H- a^  =  Va. 


6  a 


49.  V^^-^ -^ =  — Vx-b. 

^x  —  a     Va;  —  b 

50.  V6  +  a;  —  V6  —  x  =  V&. 

51.  4a;(Va  — «)  =  a  — 6. 


QUADRATIC  EQUATIONS 

52.    Vx  -\-a^-\-  ^x  —  2d^  =  V3  x. 


121 


53.    V3a^-4aa;  +  l=2(a-l). 


54.  ^/2a^x^  —  6ax  —  a^-\-5  =  a  —  l. 

55.  Va;  —  a  —  Vcia;  =  Va;  +  a. 

56.  Vax  —b  —  -Vax  +b  =  ■\/arx  -\-2ax  —  (ib. 


III.     EQUATIONS  IN  THE   QUADRATIC   FORM 
Exercise  53 

Find  all  of  the  values  of  x : 
1.    a5^-13a.-2H-36  =  0.  15.    3a;*  +  4a;^  =  4. 


2.  4a;^-29a;2  +  25  =  0. 

3.  9a;^-28a^  +  3  =  0. 

4.  9a;^+29aj2  =  80. 

5.  aj4-16  =  0. 

6.  0^4-8  =  0. 

7.  a^-a;  =  0. 

8.  2«*-a^  =  15. 

9.  x^  —  x  =  (}. 

10.  a;^  +  64a;  =  0. 

11.  a;^  =  7a^  +  8. 

12.  8a^  =  27. 

13.  a;  +  4Va;  =  5. 

14.  2  a;^  —  5  a.'^  =  3. 


16.  9a;3_37a;^  +  4  =  0. 

17.  3a;4-5a;*  =  12. 

18.  12  a;^- 11  ^  =  15. 

19.  a;«-7ar^  =  8. 

20.  4a;^-17A/^  +  4  =  0. 

21.  a;'' 4- 26^/^  =  27. 

22.  a;^  +  a^  + 1  =  0. 

23.  a;-i  +  a;"^  -  6  =  0. 

24.  2x-'-5x-'=:12. 

25.  Sx~^-^7x-^  =  6. 


26.    2</x- 


7  a5-3  =  4. 


27.  9x-^  +  4  =  37-J/^l 

28.  2.T~^-5v^^'  =  3. 


122  QUADRATIC  EQUATIONS 

29.  8-s/F^  =  15^^3  +  2.  36.  V3^^-3^/3"^^  =  10. 

30.  3x-^  =  U-19VsF'\  37.  2V5a;  +  l4-^5a;+l  =  6. 

31.  4a;-^  +  4a;-^  =  3.  38.  ^2^31 4. 3 -^2 a; - 1  =  4. 

32.  (x  +  iy  =  3(x  +  l)-\-A0,  39.  2(a;  +  l)*-3(a;  + 1)^  =  2. 

33.  (x'+3xy-2(x'+3x)=S.  40.  (4  »  + 3)^-^/4^+3  =  6. 

34.  (a^-4a;)2-9(a^-4a;)=36.  41.  3 V3^+l  +  -</3^+l=14. 

35.  (a;-l)*-13(a;-l)24-36=0.  42.  2(2a;+3)-6V2a;  +  3  =  3. 


43.   2a^  +  l-2V2ar^ 4-1  =  3. 


44.  ic2-a;  +  4-6Va:^-x  +  4  +  8  =  0. 

45.  (a^-xy-a^-{-x  =  30. 

46.  3ic2_4^^3y3ajj_4a._^2  =  2. 


47.    2a?  +  3x-\-V2a^-{-3x-\-7  =  5. 


48.  i»2  =  8-3a;-4Va^  +  3a;-3. 

49.  3a^-aj  =  6V3a^-aj-6  +  22. 


50.    a^  =  5x  +  10-2V3f-5x-2. 


51.  2a^  +  a;  +  5  =  5V2a;2  +  a;  +  l. 

52.  (a^-x  +  iy  =  3a^-3x-^l. 


53.  ic2-a;  +  5V2a^-5aj  +  6  =  |(a;4-ll). 

54.  V2^+9^T9+V2^^+7¥+5  =  V2. 

55.  aj(2a;-3)(2a^H-13aj  +  20)=0. 

56.  ax(x-l)(x'-{-l)(x'-S)  =  0. 

57.  (a^-x-12)(x'-hx-90)(a^  +  x-110)=0. 


SIMULTANEOUS  QUADRATICS  123 

SIMULTANEOUS   QUADRATICS 

Exercise  54 
Solve : 

1.  2y-Sx  =  7',  3a^-4a^-42/2  =  15. 

2.  0^  +  2/2  =  58;  xy  =  21. 

3.  3a^-2a^  =  24;  5a^-4/  =  44. 

4.  a?  —  2  y  =  1 ;  fl;2/  =  3- 

5.  x'  +  xy  +  f  =  lS',  a^-xy-hf  =  T. 

6.  2a;-2/  =  7;  &a^-Sy^  =  -7. 

7.  3a^-52/2^28;  3 ajy - 4 2/^ ^ g^ 

8.  2a^  +  a52/  — 2/^  =  ^;^  +  ^  — 2/^  =  1' 

9.  3a;  +  4?/  =  2;  ar^ - oji/ - 5 1/^  =  1. 

10.  a;4-3y  =  -4;  6a^  +  13 a;2/-5/  =  21 2/- 12a?  + 18. 

11.  a;H-2/  =  7;  a^  +  2/^=29. 

12.  a^  +  f=21S',  x-{-y  =  2. 

13.  a;2/  +  32/^  =  20;  a^-3icy  =  -8. 

14.  a;4-32/  +  4  =  0;  2a^-52/^  =  5. 

15.  2a;-32/  =  3;  4:0^-15-7 xy  =  0. 

16.  a^-a^  +  2/^=21;  3^  +  2/^  =  189. 

17.  3a^-52/2  =  7;  4:Xy-y^  =  7. 

18.  6a;-82/  +  23  =  0;  32/^-5a^-2a;  =  26. 

19.  a;-2/  =  l;  a^-2/3^i^ 


124  SIMULTANEOUS   QUADRATICS 

20.  2x'  +  3xy-4.f-  =  10;  7x~5y  =  9. 

21.  ar^_3/  =  l|5  2x2-j-/  =  4f. 

22.  1  =  ?.   l_l  =  l 

23.  a;2_^^_j_^2^3^.  ^_^^^^,^ 

24.  (^  +  2/)'-5(x-}-?/)=36;  9a;-42/  =  29. 

26.  3a^  +  22/  =  13;  a^z/2_^y_30 

27.  3a;2  +  52/-  =  17;  Aa^-3y'=lS. 

28.  0^  +  2/2=62-3^-2/;  a:?/=14. 

29.  2a^-3aJ2/  +  4/=6;  x'-^Sf  =  7. 

31.  a^H2/'=626;  x-^y  =  6. 

32.  2a;2  +  a;y/_32/2  =  8;  a;2_2^2_7 

3^  3x2-5/-3.T-22/  =  9.;  2x~3y  =  l. 

34.  0.-^  + 2/5  =  1056;  a;  +  2/  =  6. 

35.  a;-i_2/-i  =  l;    a;-3  _  2/-3  ^  3  j^ 

36.  a;  +  a^2/  =  2;  2/  +  a?2/  =  4i-. 

37.  3x'-{-2xy-2y'  =  6',  2x'  +  xy~3f  =  3. 
--38.  0^-2/2  =  16;  a^  +  2aJ2/  =  4-2/2. 

39.  xy  =  l^-  2x-5y  =  2. 

40.  a;2_|_^2^^^^^20;  x  +  y  =  3. 

41.  -1 L_  =  _44.   ?  .  4_J^ 

«-2/      0^4-2/  ^  '   x^y      xy 


SIMULTANEOUS   QUADRATICS  125 

42.  --^  =  -2|;  2x-^3y  =  2. 
y     X 

43.  2x  +  y  +  2xy  =  5;  x-\-3y +  2xy  =  l, 

44.  2x  +  3y  =  10;  23^f  +  ^5xy  =  72. 

45.  a?2_^X2/  +  /  =  '^;  ^-^2/  +  /  =  19. 

46.  5a^-2/'  =  ll;  3a5?/  +  /  =  -9. 

47.  a^/  +  14a;2/  +  24  =  0;  Sx  +  y  =  5. 

48.  a; 4-2/ =  4;  aj*  +  2/^  =  82. 

49.  x^  +  f  =  S7)  x'y-{-xy'  =  -12. 

50.  a^  +  i/2-5aj  +  52/  =  30;  a;2/  =  8. 

51.  x'  +  xy  +  f  =  19;  x' +  xV -]- y' =  931. 

1     1      ^      1   I   1      fti 

52.  i_±  =  4;   -  +  -  =  8|. 

a;     2/  aj2     2/ 

53.  x-V^  +  2/  =  9;   ^2  _^  0^2/ +  2/'  =  18^- 

54.  a^4-2a^2/  +  32/'  =  ti'  +  262;   x  +  2/  =  «- 

55.  o^  +  y^-{-x-y  =  32',   xy  =  10. 

56.  0^*4-2/^  =  5;   a;*2/^  =  6. 

57.  x^y=-117',    Vx+-y/y  =  3. 

58.  aj^-22/^  =  l;   x-Sy  =  19. 

59.  x2  4-a;2/4-2/'  =  '^5  a^'  +  «'y  +  2/'  =  91.- 

60.  a; 4- 2/ =  3;   x^  +  y^  =  33. 

61.  a;2  4-42/'-x-22/-42  =  0;   a;2/  =  12. 
e2.   a^-f  =  m',   x^y-xy^  =  30. 

63.  a;-2/  =  2;    Vac4-V2/  =  2. 

64.  x  +  y  =  13  +  V^)  a^^f  =  273-xy. 


126  PBOPERTIES  OF  QUADRATICS 

65.  J^+2Jl  =  3;   x  +  y  =  5. 

^y         ^x 

66.  2a;  +  Vi^=12;   2/4-V^=l8. 
en,    x^  +  xy-\-y'^  =  3?  —  xy-\-y'^  =  l. 

68.    x^  -\- y^  =^  xy  =  1.  69.    or  —  y"^  —  xy  =  x-{-y. 

IQ.    x^  +  y'^  =  Zxy-l;   x^-^y^  =  ll. 

71.  ^  +  ^  =  _26:   1^  +  51^=1. 
2a      6  X  y 

72.  (a;-22/)2-a;  +  22/  =  6;   3a;-52/  =  ll. 

73.  a;y  +  a;  +  2/  =  7;   o^ (a;  +  2/)  =  12. 

74.  2a:2_^3y2^8.   2(a;- l)2  +  3(2/  +  l)2  =  5. 
76.   x^^^/'^^aj^/^ig.   x-\-y  =  xy-7, 

PROPERTIES   OF   QUADRATICS 

Exercise  55 

Form  the  quadratic  equations  which  will  have  the  following 
roots ; 


1. 

7,5. 

7. 

0,  5. 

2. 

2,  ^. 

8. 

-2i,  0. 

3. 

6,-4. 

9. 

a,-l. 

4. 

4,  -If 

10. 

7,-7. 

5. 

if 

11. 

V5,  -V5. 

6. 

-1,  i- 

12. 

V-3,  -V 

3. 


13. 

a,  a  — 1. 

14. 

3+a,  -3-2a. 

15. 

2.1-3«. 

16. 

±10. 

17. 

±\/a-l. 

18. 

1  +  V2,  1-V2. 

PROPERTIES  OF  QUADRATICS 


127 


19.  3±V2. 

20.  ±Vll-5. 

7±V70 


21 


22. 


2 
3±V3 


23. 


24. 


-7±  V5 


a±  Va'-l 


25.    5±V-1. 


26. 


27. 


28. 


29. 


30. 


5±  V^r2 


-7±2V-1 


2 
5±3V^ 


Without  finding  the  actual  values  of  x,  tell  what  the  sum  of 
the  roots  is ;  their  product ;  their  character : 


31.  x^-5x-24.  =  0. 

32.  a^  +  5a;-l  =  0. 

33.  2a^-3a;-f  1  =  0. 

34.  3a^H-«-10  =  0. 

35.  x^-Sx-^5  =  0. 

36.  5a^-6x-\-2  =  0. 

37.  4a^+4a;  +  l  =  0. 

38.  4ar2  =  _a;-f4. 

39.  9x^  +  1  =  6x. 

40.  12af-\-7x  =  -6. 


41.  4a^-3a;  =  0. 

42.  4a^  =  7. 

43.  x'^-x  =  l. 

44.  3x^-i-5x-}-3  =  0. 

45.  25ar^  =  10a;-l. 

46.  3a^4-5a;  =  0. 

47.  3a^  +  5  =  0. 

48.  2a^-a;  =  l. 

49.  16a^-40a;  =  -25. 

50.  7a^  +  13ic  =  5. 


Find  the  values  of  k  which  will  make  the  following  equations 
have  equal  roots : 


51.  2x^-2x-\-k  =  0. 

52.  ko(^-4:X-\-S  =  0. 

53.  x^-\-x  =  —  k. 


54.  kx^=:3x-2. 

55.  Sx^  +  2x  =  l-k. 

56.  A;a^-A:a;  +  1  =  0. 


128  PBOPERTIES  OF  QUADRATICS 

57.  5a^  =  4a;-2A;4-l.  61.  4.x'  =  kx-k-5. 

58.  a:^-kx-^9  =  0.  62.  lla:^-\-l  =  3x-kx^-\-kx. 

59.  kx^-i-kx  =  -Sx-9.  63.  ka^-kx=7  x^-j-9  x-25. 

60.  ic2  +  49  =  A'x  +  3a;.  64.  3feic2+6A;=5a;(A;4-3)-7. 

Resolve  into  factors : 

65.  a^-3a;  +  l.         68.  a^  +  4.  71.  a^-Saft  +  fe^. 

66.  x^-x-3.  69.  a^  +  ic  +  l.  72.  17-8a;  +  a^. 

67.  Sx^-2x-2.        70.  x^-lxy-y\  73.  5a^  +  8a;-2. 

74.  Explain  the  rules  for  determining  whether  the  roots  of 
an  equation  are  real  or  imaginary.  Equal  or  unequal.  Rational 
or  irrational. 

75.  If  the  sum  of  the  roots  of  a  quadratic  is  3  and  their 
product  is  2,  find  the  difference  of  the  roots.  Find  the  differ- 
ence of  the  squares  of  the  roots.  Find  the  sum  of  the  recipro- 
cals of  the  roots. 

76.  Find  the  condition  that  one  root  of  ax^  -\-'bx-\-c  =  0 
shall  be  the  reciprocal  of  the  other.  Find  the  condition  that 
one  root  shall  be  double  the  other.     One  three  times  the  other. 

77.  If  m  and  n  stand  for  the  roots  of  2  ic^  -f  5  .'c  —  3  =  0,  find 

the  values  of :     (a)  m  +  n.    (c)  m  —  n.       (e)  — | 

^  m     n 

(b)  mn.         (d)  m^  —  n\     (/)  m^-\-7i\ 

78.  Find  the  values  of  the  same  expressions  in  the  equation 
3  a:^  =  13  a;  + 10.     Also  in  equation  3  .t^  —  a^  + 1  =  0. 

79.  Form  the  quadratic  whose  roots  shall  be  |  and  |.  Form 
that  whose  roots  shall  be  |  and  |.     Compare  the  results. 


hatio  and  propohtion  129 

RATIO   AND   PROPORTION 
Exercise  56 

1.  Find  a  mean  proportional  between  5|  and  27.    Between 

m  and  n^.     Between  -—  and      '  ^» 

a  a 

2.  Find  a  fourth  proportional  to  3,  5,  12.     To  a,  a  +  1,  a^. 
To  6,  8,  lOf     To  8,  lOi  6. 

3.  Find  a  third  proportional  to  4  and  10.     To  3  and  3|. 
To  a  and  a;  - 1.     To  i  and  |. 

4.  Solve  2a;-l:3a;-2  =  3(a;  +  l):5a;  +  l. 

5.  Solve  a;-5:3a;-fl  =  5-8a;:3(l-2a;). 

6.  Solve    l:l  =  -l_:i. 

a    6     a—b    or 

7.  Solve    — :  -— ! — -  =  x:a-\- 


c{a-\-c)    a^  —  c^  a  —  c 

8.    Solve 
2x^-3x  +  l:3x'-3x-^l  =  Say'-2x-5:4:x'-2x-5. 
2a^-4:X-l        x'-\-x-2 


9.    Solve 


2a^-f2a;-l      a^+13a;-2 


10.  Solve   .^  +  3a.-7^a^  +  4a.  +  10, 

a^  — 5a;4-6       a:^  — 4a;  +  4 

11.  Solve  ^^-^^-^==^  +  ^  +  ^. 

12.  Solve   ^-2.^-4-2     2^  +  . ^1 

a^_3ar^  +  2     2x2-x-l 


13.    Solve   V«  +  4  :  Va;  - 1  =  V6  a;  +  6  :  V5  a;  —  9. 


14.    Solve  V3a;-2:V4a;  +  l  =  V7a;  +  2:2V5aj-l. 

R.   &  S.  EX.   IN  ALG.  —  9 


130  RATIO  AND  PROPORTION 

15.  Two  numbers  are  in  the  ratio*  of  3 : 7  and  their  sum 
is  60.     Find  them. 

16.  Three  numbers  are  in  the  ratio  of  2  :  3  :  4  and  their  sum 
is  63.     Find  them. 

17.  Find  two  numbers  in  the  ratio  of  2  :  5,  the  sum  of  whose 
squares  is  464. 

18.  Find  three  numbers  in  the  ratio  of  1:2:3,  the  sum  of 
whose  squares  is  126. 

19.  What  number  added  to  each  of  the  numbers  2,  5,  11,  15 
will  make  the  sums  proportional  ?  • 

20.  Find  a  mean  proportional  and  a  third  proportional  to  5 
and  20.     Also  to  3i  and  H. 

21.  It  a:  b  =  c:d,  prove  the  property  of  " composition '^  by 
use  of  the  equivalents,  a  =  bx  and  c  =  dx.  Prove  "  division  " 
by  the  same  method. 

22.  If  a:b  =  c:d  =  e  :f=g :  h,  prove  by  the  method  of 
example  21  that  a-\-c-\-e  +  g'.h-\-d 4-/4- h  =  a:b=c:  d  =  etc. 

23.  If  a:  b  =  c:  d,  prove  that  a4-3c:  b-^Sd  =  2  a-\-c:  2b-\-d. 

24.  If  m  :  n=p  :  q,  prove  that  m-^n  : p-\-q  =  m  —  2  n  : p—2  q. 

25.  It  X  :  y  =  z  :  w,  prove  that 

x^ -\- y^ :  z^  +  iv^  =  (x  —  nyy :  (z  —  7iwf. 

26.  It  p  :  q  =  r  :  s,  prove  that 

Vi>^  4-  7^  :  Vg^  -\-s^  =  ap  —  br:aq  —  bs=p:q. 

27.  It  a:b:  :b  :c,  prove  that  a-{-Sb:b-\-Sc  =  a:b  by  use 
of  the  equivalents  a  =  cx^  and  b  =  ex. 

28.  If  2/  is  a  mean  proportional  between  x  and  Zj  prove  that 

x-2y:y-2z  =  2x-'^y\2y-^z. 


RATIO  AND  PROPORTION  131 

29.  If  a,  b,  c,  d  are  in  continued  proportion,  prove  by  use 
of  the  equivalents  a  =  da?,  b  =  daf,  G  =  dx  that  a  +  6  +  c  :  a  +  & 
=  6  +  cH-cZ:54-c. 

30.  If  a,  b,  c,  d  are  in  continued  proportion,  prove  that 

a-\-b^:c-\-c^  =  b-{-c':d-\-d\ 

31.  If  a,  b,  c  are  in  continued  proportion,  prove  that 

a  +  b  :b  +  c  =  b^:  ac^. 

32.  If  a,  b,  c,  d  are  in  continued  proportion,  prove  that 

a^^b' -{-(?:  b'  +  c'-\-d'={a  +  c){a-c):(b-\-d){b-d)  =  a':b^ 

33.  If  _^_  =  _L=_!_,  prove  that  x-y-{-z  =  0. 

b  +  c     a-\-c     a  —  b 

34.  If  _l_=_!^  =  _i!_,  prove  that  l  +  m  +  n  =  0. 

b  —  c     c  —  a     a  —  b 

35.  If  a  +  2b-\-c:b-\-c  =  a-{-b:b,  prove  that  6  is  a  mean 
proportional  between  a  and  c. 

36.  Find  two  numbers  in  the  ratio  of  2 :  3  such  that  the 
sum  of  their  squares  is  to  their  product  increased  by  2,  as  2  : 1. 

37.  If  1  be  added  to  each  of  two  numbers,  their  ratio  is  1 :  2. 
The  difference  of  their  squares  is  to  3  more  than  their  product 
as  5:3.     Find  them. 

38.  There  are  two  numbers  such  that  the  ratio  of  the  sum 
of  their  cubes  and  the  cube  of  their  sum  is  7  : 1 ;  and  if  6  be 
added  to  each,  the  ratio  of  these  sums  is  1 :  4.     Find  them. 

39.  For  what  value  of  x  will  2  a;  —  1  be  a  mean  proportional 
between  x-\-5  and  4  a;  —  13  ? 

40.  What  values  must  x  have  in  order  that  2  a;  —  7,  3  a;  + 1, 
4  a;  —  3,  5  (a;  + 1)  may  form  a  true  proportion  ? 


132  VARIATION 

VARIATION 

Exercise  57 

1.  li  X  varies  as  y  and  y  =  2  when  x  =  12,  find  x  when  y  =  ^. 

2.  It  xccy-  and  x  =  ^  when  y  =  ^,  find  ?/  when  a;  =  18. 

3.  If  A  varies  inversely  as  B  and  A  =  —  6  when  ^  =  —  i  find 
^  when  -B  =  |. 

4.  If  ^  varies  jointly  as  B  and  C  and  ^  =  9  when  5  =  0  =  6, 
find  A  when  B  =  5  and  O  =  —  8. 

5.  If  07  varies  directly  as  y  and  inversely  as  z,  and  a;  =  2 
when  y  =  3  and  2  =  6,  find  «/  when  x  =  S  and  2;  =  —  3. 

6.  li  xccy  and  a;  is  3  when  y  =  -|,  find  an  equation  between 
X  and  y. 

7.  If  ic  X  -  and  ?/  =  —  5  when  a;  =  2,  find  the  equation  joining 
X  and  ?/. 

2/ 

8.  If  a;  X  -  and  a;  =  15  when  y  =  5  and  2;  =  4,  find  a;  in  terms 

z 

of  2/  when  2  is  —  1. 

9.  If  a!  X  (2y  +  5)  and  a;  =  3  when  ?/  =  —  2,  find  y  if  x  =  6. 

10.  Given  that  ?/-  x  (a;^  +  1)  and  a;  =  7,  when  j^  =  10,  find  x 
when  2/  =  VTO- 

11.  If  u  is  equal  to  the  sum  of  two  quantities,  one  of  which 
varies  as  x  and  the  other  inversely  as  x,  and  if  u  =  —  l  when 
a;  =  |,  and  w  =  1  when  a;  =  1,  find  the  equation  between  u 
and  X. 

12.  If  V  is  equal  to  the  sura  of  two  quantities,  one  of  which 
varies  as  a^  and  the  other  inversely  as  1/,  and  v  =  —  1,  when 
X  =^,  y  =  2',  and  v  =  7  when  x  =  2,  y  =  3 ;  find  the  equation 
for  V  in  terms  of  x  when  ?/  =  —  1. 


VABIATION  133 

13.  Given  that  y  =  the  sum  of  three  quantities  which  vary- 
as  X,  x"^,  and  x^  respectively.  When  x=l,  ?/=4;  when  x=2, 
y  =  S\  when  x  =  3,  y  =  IS.     Express  y  in  terms  of  x. 

14.  If  y  varies  inversely  as  ar^  —  1  and  y  =  —  5  when  a;  =  4, 
find  X  when  y  =  —  15. 

15.  If  y  varies  inversely  as  (2  x  +  1)  (x  —  3)  and  y  =  —  \ 
when  a;  =  2,  find  a;  when  y  —  1\. 

16.  If  the  area  of  a  circle  varies  as  the  square  of  its  radius, 
and  the  area  of  a  circle  whose  radius  is  7  is  154,  find  the  area 
of  the  circle  whose  radius  is  10. 

17.  Find  the  radius  of  the  circle  equivalent  to  the  sum  of 
two  circles  whose  radii  are  5  and  12  respectively. 

18.  The  pressure  of  the  wind  upon  a  plane  surface  varies 
jointly  as  the  area  of  the  surface  and  the  square  of  the  wind's 
velocity.  The  pressure  on  a  square  foot  is  1  pound  when  the 
wind  is  blowing  at  the  rate  of  15  miles  per  hour.  Find  the 
velocity  of  the  wind  when  the  pressure  on  a  square  yard  is 
36  pounds. 

19.  If  w  varies  as  the  sum  of  x,  y,  and  z,  and  tv  =  3  when  a;  =  3, 
y  =  —  4:,z=6,  find  xiiw  =  —  3,y  =  3^,z  =  —  9. 

20.  If  w  is  equal  to  the  sum  of  two  quantities,  one  of  which 
varies  as  x,  and  the  other  jointly  as  y  and  z,  and  w  =  —  3  when 
a;  =  2,  2/  =  6,  2  =  —  1;  and  w  =  —  2  when  a;  =  4,  ?/  =  2,  2;  =  —  3  ; 
find  the  equation  combining  the  four  quantities,  w,  x,  y,  and  z. 

21.  If  the  square  of  x  varies  as  the  cube  of  y,  and  a;  =  3 
when  y  =  2,  find  y  when  x  =  24. 

22.  The  area  of  a  triangle  varies  jointly  as  its  base  and 
altitude.  Find  the  altitude  of  a  triangle  whose  base  is  23, 
equivalent  to  the  sum  of  two  triangles  whose  bases  are  15  and 
22  and  whose  altitudes  are  10  and  12  respectively. 


134  ARITHMETICAL  PROGRESSION 

ARITHMETICAL   PROGRESSION 

Hzercise  58 

In  the  following  16  examples  tell  what  a  is,  what  d  is,  what 
n  is.     Also  find  I  and  s  in  each. 

1.  5,  7,  9,  •••,  to  15  terms. 

2.  6,  9,  12,  ...,  to  10  terms. 

3.  —  2,  —  31  —  5,  •••,  to  45  terms. 

4.  3,  3.1,  3.2,  ...,  to  300  terms. 

5.  8,  7.5,  7,  •••,  to  60  terms. 

6.  2|,  2^,21,  ...,  to55terms. 

7.  -  3^,  -  2f,  -  21   .-.,  to  75  terms. 

8.  1  +  a;,  1  +  3  a;,  1  +  5  X,  •  •  •,  to  10  terms. 

9.  Odd  numbers  to  37  terms. 

10.  Numbers  divisible  by  7  to  15  terms. 

11.  Numbers  divisible  by  3  to  20  terms. 

12.  5,  10,  15,  •••,  to  r  terms. 

13.  1,  2,  3,  4,  •••,  to  X  terms. 

14.  2,  6,  10,  14,  ...,  to  w  terms.      . 

15.  The  first  n  odd  numbers. 

16.  The  first  2  71  even  numbers. 
Insert,  between 

17.  11  and  32,  5  arithmetical  means. 

18.  7^  and  30,  9  arithmetical  means. 


ARITHMETICAL   PROGRESSION  135 

19.  38|  aud  —  44|,  99  arithmetical  means. 

20.  17  and  3,  12  arithmetical  means. 

Find  d  and  s  if : 

21.  a  =  5,  Z  =  25,  n  =  ll.  23.    a  =  4,  Z  =  36,  n  =  24. 

22.  a  =  -13,  Z  =  26,  n  =  14.       24.    a  =  12i,  ^  =  - 13|,  n  =  40. 

Find  n  and  s  if : 

25.  a  =  6,d  =  2,lz=S0.  27.   a  =  3J,  c?  =  J,  ;  =  lOf 

26.  a  =  -17,  d  =  4,  ^  =  39.  28.    a=9i,  d  =  -i,  l  =  -W^. 

Find  a  and  s  if : 

29.  d  =  3,  ^  =  38,  n  =  ll.  31.    d=-2,l=-25,n  =  27. 

30.  (7  =  1|,  Z  =  69,  n  =  41.  32.    d  = -|,  Z  =  6^,  n  =  20. 

Find  Z  and  d  if : 
33.    a  =  5,  n  =  9,  s  =  297.  34.    a  =  3J,  n  =  15,  s  =  78|. 

35.    a  =  -l|,  n  =  30,  s  =  530. 

Find  n  and  d  if : 

36.  a  =  8,  ^  =  41,  s  =  294.  38.    a  =  8,  Z  =  0,  s=  100. 

37.  a  =  3i  Z  =  42|,  s  =  621.        39.  a=-3^,Z=-36,s=-790. 

Find  a  and  /,  if : 
40.    d  =  S,  n  =  13,  s  =  260.         41.    d  =  i,  n  =  20,  s  =  102|, 
42.    d  =  -f,  n  =  8o,  s  =  -306i. 

Find  a  and  d,  if : 
43.    Z  =  47,  n  =  23,  s  =  575.        44.    Z  =  ll|,  n  =  37,  s  =  209J. 
45.    /  =  -16^,  n  =  43,  s  =  43. 


136  ARITHMETICAL  PROGRESSION 

Find  n  and  ?,  if : 
46.    a  =  S,  d  =  2,  s  =  80.  47.    a  =  2,  d  =  -S,  s  =  -328. 

48.    a  =  o,  d  =  —  ^,  s  =  27. 
Find  n  and  a,  if : 

49.  d  =  5,   l  =  S2,  5  =  119.  51.    d  =  l,  1  =  6,  s  =  45. 

50.  d  =  -^,  /  =  5i    s  =  2o.        52.    fZ  =  -|,  /  =  -3,  s  =  13. 

53.  How  many  numbers  are  there  between  100  and  1000 
that  are  exactly  divisible  by  7  ?     Find  their  sum. 

54.  Find  the  sum  of  all  the  numbers  of  two  figures  each 
that  are  divisible  by  8. 

55.  Find  the  sum  of  the  first  50  odd  numbers. 

56.  In  the  series  2,  5,  8,  •••,  which  term  is  98  ? 

57.  How  many  terms  must  be  taken  from  the  series  3,  5, 
7,  •••,  to  make  a  total  of  255  ? 

58.  Which  term  of  the  series  li  2,  21  •••,  is  24?  How 
many  consecutive  terms  must  be  taken  from  this  series  to 
make  84? 

59.  The  7th  term  of  an  A.  P.  is  17,  and  the  12th  term  is  27. 
Find  the  1st  term.     The  3d  term. 

60.  The  10th  term  of  an  A.  P.  is  |,  and  the  J  8th  is  3f . 
Find  the  1st  term.     The  100th  term.     Sum  of  20  terms. 

61.  How  is  a  single  arithmetical  mean  between  2  numbers 
found  most  readily  ?  How  do  you  determine  whether  or  not 
3  numbers  are  in  A.  P.  ? 

62.  Find  x,  so  that  3  —  5x,l-\-2x,4:-\-7x,  shall  form  an  A.  P. 

63.  The  sum  of  4  numbers  in  A.  P.  is  46,  and  the  product  of 
the  2d  and  3d  is  130.     Find  them. 

64.  The  sum  of  3  numbers  in  A.  P.  is  27,  and  the  sum  of 
their  squares  is  275.     Find  them. 


ARITHMETICAL  PROGRESSION  137 

65.  A  body  freely  falling  from  a  position  of  rest  will  fall 
16 J^  feet  the  first  second,  48^  feet  the  second  second,  SOy^  feet 
the  third,  and  so  on.  Find  the  distance  fallen  during  the  10th 
second.  How  far  in  10  seconds?  How  far  in  20th  second? 
How  far  in  20  seconds  ? 

66.  Find  x,  so  that  S  -{-  2  x^  5  -{-  6  x,9  +  5  x,  shall  form  an  A.  P. 

67.  Which  term  of  the  series  2^,  3f,  5,  •••,  is  45  ? 

68.  How  many  consecutive  terms  in  the  series  2^,  3|,  5,  •••, 
will  make  67^  ?     Interpret  the  negative  result. 

69.  If  the  6th  term  of  an  A.  P.  is  9  and  the  16th  term  is 
22J,  find  the  25th  term  and  the  sum  of  30  terms. 

70.  Find  the  sum  of  the  series  x,  Sx,  5x,  7 x,  ••.,  to  x  terms. 

71.  Find  the  sum  of  all  the  numbers  between  100  and  600 
that  are  divisible  by  11. 

72.  Find  x,  so  that  2a;  —  1,  3a;  +  2,  6aj  +  8,  shall  be  an  A.  P. 

73.  What  will  x  and  y  each  be,  if  the  four  terms  2x  —  y, 
x-\-2y,3x  +  y^7x  —  10,  form  an  A.  P.  ? 

74.  Find  the  sum  of  15  terms  of  an  A.  P.  of  which  the 
middle  one  is  lOJ. 

75.  Find  the  sum  of  '1±1 -f- !?i±^  +  ^?i±^ . . .  to  n  terms. 

n  n  n 

76.  A  boy  travels  at  the  rate  of  1  mile  the  first  day,  2  the 
second,  3  the  third,  and  so  on;  6  days  later  a  man  sets  out 
from  the  same  place  to  overtake  him,  traveling  15  miles  every 
day.  How  many  days  must  elapse  after  the  second  starts 
before  they  are  together?     Interpret  both  results. 

77.  The  sum  of  n  terms  of  the  series  21,  18,  15,  •••,  is  equal 
to  the  sum  of  the  same  number  of  terms  of  the  series  3,  3^, 
3_6_,  ....     Find  n. 

78.  Find  the  sum  of  41  terms  of  an  A.  P.  whose  21st  term  is 
100. 


138  GEOMETRICAL   PROGRESSION 

GEOMETRICAL  PROGRESSION 

Exercise  59 

Find  I  and  s  in  each : 

1.  3,  6,  12,  •..,  to  8  terms. 

2.  2,  8,  32,  ...,  to  5  terms. 

3.  40,  20,  10,  ...,  to  6  terms. 

4.  2.1,  21,  210,  ...,  to  5  terms. 

5.  54,  18,  6,  ...,  to  5  terms. 

6.  3.2,  0.32,  .032,  ...,  to  6  terms. 
'^'    ^j  f?  \h  •••?  to  5  terms. 

8.  I,  4^2,  ...,  to  7  terms. 

9.  11   —3,  6,  ...,  to  9  terms. 

10.  -  5,  15,  -  45,  . . .,  to  5  terms. 

11.  34,  If,  I,...,  to  10  terms. 

12.  16J,  -111  71  ...,  to  5  terms. 

13.  l+a;  +  «2  +  a^---,  to6  terms. 

14.  32-16  +  8-4  +  2-1...,  toTi  terms. 
Find  r  and  s,  if : 

15.    a  =  3,     ^  =  48,     w  =  5.  16.    a=^n,     1  =  4.05,     n  =  5 

17.    a  =  131       Z  =  17,      ^::=a 
Find  a  and  s,  if : 
18.    Z=i,     71  =  6,     r=i-.  19.    /  =  85i      n  =  5,     r  =  lj. 

20.    Z  =  |,     71  =  5,     r  =  -2 


GEOMETBICAL  PROGRESSION  139 

Find  n  and  s,  if  : 

21.  a  =  5,     1  =  160,     r  =  2.        23.   a  =  24,  /=|,        r  =  f 

22.  a=3,     Z  =  1875,  r  =  5.        24.    a  =  f,     Z  =  -24,   r  =  -2. 

Find  r  and  r?.,  if  : 

25.  a  =  2,    Z  =  486,s  =  728.     27.  a  =  1|,  Z  =  135,  s=  201f. 

26.  a  =  56,  Z  =  lf,    8  =  1101    28.  a  =  |,  Z  =  -  ^^^ ^  s  =  - 8|f f . 

Insert,  between 

29.  4    and      972,   4  geometrical  means. 

30.  7    and      896,   6  geometrical  means. 

31.  5^  and      40 J,   4  geometrical  means. 

32.  20and— yl-g^,   8  geometrical  means. 

33.  7^  and        ff,   4  geometrical  means. 

Find  the  sum  of  each  series  to  infinity  : 

34.  6,3,  H,....  38.    8|,  -6|,5,-... 

35.  1,  -|,  1  ....  39.   8.3,  0.83,  .083, .... 

36.  15,  5,  If,  ....  40.    .72,  .0072,  .000072,  .... 

37.  18,12,8,....  41.    1^,0.75,0.5.... 

42.  0.4545,  ....  44.   3.8181,  ....  46.    2.34848,  .... 

43.  0.05454,....  45.    5.12727,....  47.    1.026363,.... 

48.  If  the  3d  term  of  a  G.  P.  is  36  and  the  6th  term  is  972, 
find  the  1st  and  2d  terms. 

49.  If  the  4th  term  is  24  and  the  8th  term  is  384,  find  the 
first  2  terms. 

50.  The  3d  term  is  4  and  the  7th  is  20^.     Find  the  first 
2  terms. 


140  GEOMETRICAL  PROGRESSION 

51.  In  the  G.  P.  2,  6,  18,  .••,  which  term  is  486  ? 

52.  How  many  terms  must  be  taken  from  the  series  9,  18, 
36,  •••,  to  make  a  sum  of  567  ? 

53.  How  many  consecutive  terms  in  the  series  48, 24, 12,  •••, 
are  required  to  make  95 J  ? 

54.  The  1st  term  of  a  G.  P.  is  8.  Its  sum  to  infinity  is  32. 
Find  the  ratio. 

55.  How  can  a  single  geometric  mean  be  determined  most 
readily  ?  How  does  one  test  a  series  to  determine  whether  it 
is  a  G.  P.  or  not  ? 

56.  Find  ic,  if  2  a;  —  4,  5  a;  —  7,  10  a;  +  4,  are  in  G.  P. 

57.  There  are  3  numbers  in  A.  P.  whose  common  difference 
is  4.  If  2,  3,  9,  be  added  to  them  respectively,  the  sums  form 
a  G.  P.     Find  the  numbers. 

58.  The  sum  of  a  G.  P.  to  infinity  is  18  and  the  2d  term  is  4. 
Find  the  1st  term  and  ratio. 

59.  If  the  series  a;  -f  1,  x-\-S,4:X  —  3,  is  geometric,  find  x. 
Find  a;  if  it  is  an  A.  P.  Find  the  4th  term  of  the  series  in 
each  case. 

60.  Tell  whether  each  of  the  following  series  is  arithmetical 
or  geometrical : 

(a)   3,  6,  12,  ....  (c)   12,  18,  25,  .... 

(6)   6,  12,  18,  ....  (d)  3i,  H,  0.6,  .... 

61.  The  sum  of  three  numbers  in  G.  P.  is  65.  The  sum  of 
the  first  two  is  i  the  sum  of  the  last  two.     Find  them. 

62.  Divide  49  into  3  parts  in  G.  P.  such  that  the  sum  of 
the  1st  and  3d  parts  is  2i  times  the  middle  part. 

63.  The  sum  of  3  numbers  in  G.  P.  is  14  and  the  sum  of 
their  reciprocals  is  f .     Find  them. 


GEOMETRICAL  PROGRESSION  141 

64.  Insert  between  6  and  16  two  numbers,  such  that  the 
first  three  of  the  four  shall  be  in  A.  P.  and  the  last  three 
in  G.  P. 

65.  If  the  series  3^,  2|^,  •••,  be  an  A.  P.,  find  the  105th  term. 
If  a  G.  P.J  find  the  sum  to  infinity. 

66.  The  sum  of  $  240  was  divided  among  4  men  in  such^a 
way  that  the  shares  were  in  G.  P.,  and  the  difference  between 
the  greatest  and  least  shares  is  to  the  difference  between  the 
other  two,  as  13  :  3.     Find  each  share. 

67.  What  number  added  to  each  of  the  numbers  2,  5,  11, 
will  make  sums  that  are  in  G.  P.  ? 

68.  Find  x,  so  that  5-\-x,  5  —  x,  2(1  —  5  a;),  shall  be  in  G.  P. 

69.  If  4  a?  —  1,  6  a;  +  1 ,  5(2  x  -f- 1),  are  in  G.  P.,  find  x  and  find 
the  ratio.     Also  find  the  next  term. 

70.  If  the  first  term  of  a  G.  P.  is  6  and  the  sum  to  infinity 
is  18,  find  the  third  term. 

71.  If  a  man  ascends  a  mountain  at  the  rate  of  81  yards  the 
first  hour,  54  yards  the  second,  36  yards  the  third,  etc.,  how 
many  hours  will  he  require  to  ascend  211  yards  ? 

72.  There  are  4  numbers,  the  first  three  of  which  are  in 
G.  P.,  and  the  last  three  are  in  A.  P.  The  sum  of  the  first  and 
last  is  14,  and  the  sum  of  the  second  and  third  is  12.  Find 
the  numbers. 

73.  A  ball  thrown  vertically  into  the  air  150  feet  falls  and 
rebounds  60  feet.  It  falls  again  and  rebounds  24  feet,  and  so  on 
until  it  comes  to  rest  on  the  ground.  Find  the  entire  distance 
through  which  the  ball  has  traveled. 

74.  Prove  that  equimultiples  of  a  G.  P.  are  also  in  G.  P., 
and  that  alternate  terms  of  a  G.  P.  form  another  G.  P. 


142  PERMUTATIONS  AND   COMBINATIONS 

PERMUTATIONS   AND   COMBINATIONS 
Exercise  60 

1.  How  many  numbers  of  5  different  figures  each  can  be 
formed  from  our  9  significant  digits  ? 

2.  How  many  words  of  4  letters  each  can  be  formed 
from  the  26  letters  in  our  alphabet,  no  letter  being  repeated 
in  the  same  word  ? 

3.  Find  the  number  of  committees,  each  containing  5  boys, 
that  can  be  selected  from  a  room  of  20  boys. 

4.  Find  the  number  of  combinations  of  8  objects  each  that 
can  be  formed  from  25  objects. 

5.  How  many  different  words  can  be  formed  from  the 
letters  in  the  word  TJiursday,  using  all  its  letters  each  time  ? 

6.  From  the  members  of  a  party  of  30  people,  a  board  of 
4  officers  is  to  be  chosen.     In  how  many  ways  can  this  occur  ? 

7.  From  the  letters  in  the  word  Repiihlican  how  many 
words  of  4  letters  can  be  found  ?     Of  5  letters  ?     Of  7  letters  ? 

8.  The  prime  factors  of  a  certain  number  are  2,  5,  7,  11, 
and  17.  How  many  exact  divisors  (except  itself  and  unity) 
has  this  number  ? 

9.  It  is  required  to  place  20  dissimilar  bouquets  in  the  form 
of  an  arch.     In  how  many  ways  can  they  be  arranged  ? 

10.  From  the  9  significant  digits,  how  many  numbers  can 
be  formed  each  containing  1  digit  ?  Two  different  digits  ? 
3?     4?     5?     6?     7?     8?     9?     All  together? 

11.  There  are  25  points  in  a  certain  plane;  these  are  joined 
so  as  to  form  triangles  having  the  vertices  at  the  points.  How 
many  triangles  will  there  be  ? 


PERMUTATIONS  AND  COMBINATIONS  143 

12.  From  the  letters  in  the  word  handiwork  how  many 
words  of  5  letters  can  be  formed  ?  How  many  of  these  will 
contain  the  h  ?  the  w  ?  How  many  will  begin  with  d  ?  How 
many  will  contain  both  d  and  i?  How  many  will  not  con- 
tain d  ? 

13.  A  man  has  5  pairs  of  trousers,  8  vests,  and  6  coats.  In 
how  many  different  costumes  can  he  appear  ? 

14.  Six  persons  enter  a  car  in  which  there  are  10  seats.  In 
how  many  ways  can  they  be  seated  ? 

15.  In  how  many  ways  can  a  baseball  nine  be  arranged 
provided  the  pitcher  is  always  the  same  ?  Provided  the  pitcher 
and  catcher  are  always  the  same  individuals  ? 

16.  In  how  many  ways  can  10  people  arrange  themselves 
around  a  circular  table  ? 

17.  How  many  words  can  be  formed  from  the  letters  in  the 
word  latin,  the  2d  and  4th  being  always  vowels  ? 

18.  How  many  words  can  be  formed  from  the  letters  in  the 
word  united,  the  even  places  being  always  occupied  by  con- 
sonants ? 

19.  How  many  words  can  be  formed  from  the  letters  in  the 
word  education,  provided  the  2d,  4th,  6th,  and  last  letters  are 
always  consonants  ? 

20.  From  our  9  digits  how  many  numbers  can  be  formed, 
each  containing  6  figures  ?  How  many  of  these  will  contain 
the  figure  5  ?  How  many  will  not  contain  a  7  ?  How  many 
will  contain  both  5  and  7  ?  How  many  will  begin  with  3  ? 
End  with  4  ?     How  many  will  be  odd  ? 

21.  From  10  gentlemen  and  8  ladies  how  many  committees 
can  be  chosen,  each  containing  3  gentlemen  and  2  ladies  ? 

22.  From  10  consonants  and  5  vowels  how  many  words,  each 
containing  4  consonants  and  3  vowels,  can  be  formed  ? 


144  PERMUTATIONS  AND   COMBINATIONS 

23.  There  are  8  Democrats  and  10  Kepublicans  belonging  to 
a  certain  board.  How  many  committees  can  be  chosen  each 
having  4  Democrats  and  5  Kepublicans  ? 

24.  Out  of  4  vowels  and  9  consonants  there  are  words  to  be 
constructed,  each  consisting  of  2  vowels  and  6  consonants. 
How  many  can  there  be  ? 

25.  From  6  white  balls,  4  red  balls,  and  8  black  balls,  how 
many  combinations  can  be  made  each  to  contain  3  white,  2  red, 
and  4  black  balls  ? 

26.  From  4  labials,  6  vowels,  5  palatals,  how  many  words 
can  be  made  each  consisting  of  2  labials,  3  vow^els,  and  2 
palatals  ? 

27.  How  many  different  sums  of  money  can  be  made  from 
the  following  coins:  cent,  5-cent,  dime,  quarter,  half  dollar, 
and  dollar  ? 

28.  A  guard  of  5  men  must  be  selected  every  night  out  of  a 
detachment  of  32  men.  For  how  many  nights  can  a  different 
guard  be  selected  ?     How  many  times  will  each  soldier  serve  ? 

29.  A  company  of  15  merchants,  12  lawyers,  and  8  teachers 
wish  to  form  a  commission  from  their  number,  consisting  of  4 
merchants,  3  lawyers,  and  2  teachers.  How  many  ways  are 
there  in  which  they  can  do  it  ? 

30.  Find  the  number  of  permutations  that  can  be  made  from 
the  letters  in  the  following  words  using  all  the  letters : 

(a)  Recess.  (c)  Bumblebee.  (e)  Concnn-ence. 

(b)  Possess.  (d)   Tennessee.  (/)   Unostentatious. 

31.  In  how  many  different  ways  can  one  mail  4  letters  in  a 
village  containing  7  letter  boxes  ? 

32.  How  many  different  quantities  can  be  weighed  with  the 
following  weights:  1  ounce,  3  ounces,  8  ounces,  10  ounces, 
1  pound,  5  pounds,  and  10  pounds? 


BINOMIAL   THEOHEM  145 

33.  With  2  violet,  2  indigo,  3  blue,  4  green,  1  yellow,  1  orange, 
and  2  red  flags,  how  many  signals  can  be  made  if  all  the  flags 
are  used  and  always  kept  in  a  vertical  column  ? 

34.  Prom  7  consonants  and  5  vowels  how  many  words  can 
be  formed,  each  consisting  of  4  consonants  and  2  vowels  ? 

35.  A  plane  is  determined  by  3  points,  if  they  are  not  all 
in  a  straight  line.  How  many  planes  are  determined  by  100 
points  (no  four  of  them  lying  in  the  same  plane)  ? 


BINOMIAL   THEOREM 

Exercise  61 
Expand : 

1.  (a-2y.  6.  (2m-\-ny.  '       9.  (■</6-2)«. 

2.  (2a  +  l)^  6.  (3a-Va6)*.  10.  (V2-h2V6)3. 

3.  {l-^a'xy.  7.  {a-V^-iy.  11.  (Va-Vw)'*'. 

4.  (1-2/y.  8.  (V2-V3)^  12.  (a^-^a^c/. 

13.  (1-3V3)1  20.    r^  +  ^Y- 

14.  (Vi-3V2^)*. 


21.  {a</x-'-x^a-y 

22.  (2aV2^+-v/4)*. 


15.  (V3«-^2a;2/)^ 

16.  (2-V^^y.  ,_      ,, 

^  ^  23.    (V2a-</-3ay, 

17.  (3V-1  +  2V5)'.  r2Va^3V2l^ 

24.      ^  -zzz  -\ — — 

18.  {cix~'^  —  •\/a~'^x)  •  I  Vn^         ^a 

^^'      V^-3V^=3^    .  25.       -^-^^ 
V  ^  .S  «                      /  la  V  a  V  c     J 


R.   &  S.   EX,   IN  ALG.  —  10 


146  BINOMIAL   THEOREM 

Find  only  the  term  required : 

26.  TheTth  termof  (aj  +  V^)". 

27.  The  5th  term  of  (1  -  2  xf. 

28.  The  4th  term  of  (a  +  3  ■\/xf\ 

29.  The  6th  term  of  (2  n ^/m  - 1)". 

30.  The  8th  term  of  (a  V^  -  a;  VaP. 

31 .  The  7th  term  of  (i  -  2  a;  ^^)". 

32.  The  5th  term  of  (Vox +^?y«. 

33.  The4thtermof  (V6- V3)«. 

34.  TheSth  termof  (V3  +  V-2)». 

35.  The  7th  term  of  ( V2  -  a  V3xy\ 

36.  The  9th  term  of  (J^ ^a  +  2  V3^)^. 

37.  The  4th  term  of  ( V2  -  5  V^^. 

88.   The  5th  term  of  [^  +  -^^^T- 

39.  The  term  containing  o^  in  (1  —xVxy^. 

40.  The  term  containing  x'  in  (a;-f  2V^)^. 

41.  The  term  containing  x^^  in  (■\/x—-^xyy\ 

42.  The  term  containing  x^  in  (2 \/x—  V2xy. 

43.  The  term  containing  x-*  in  f— ^—\  . 

V6a:      V^V 


BINOMIAL   THEOREM  147 

Expand  to  four  terms : 


-I      ..  1 


44.  (a  +  6)-^              52.    (x-2-^/ax^-^,     59  

3. ^                 '  Va-\-2¥ 

45.  (a-a;)-\              53.    Va^  +  Va.  o  ah 

60. 


46.  (a;-2)-^  54.    (l  +  5f(2^x)i  ^a'^  -  aft^ 

47.  (a  +  2/)i  55.    (a6«-36V«)l    ^^^    T^^^' 

48.  (a2_4)-2^  56.    5(x-^-\/2xy)-K   gg.    (a2_3V2)l 


49.  (l-2^-x)-\   57.    C-L_y3^y'^   63.    -—4 

Wa  J  VaVx 

50.  (aar«  +  l)^. 
58. 64 


ax 


2y 


51.    (b^-2by\  a-b^/c  (x-2^xyy 

Find  only  the  terra  required : 

65.  The  3d  term  of  (a  +  b)'\ 

66.  The  4th  term  of  (x  —  y)-\ 

67.  The  4th  term  of  (1  +3Va5)i 

68.  The  5th  term  of  (a;- 2 VaJ^) -2. 

69.  The  5th  term  of  {l-^/2x)-\ 

70.  The  6th  term  of  (a  +  V-2«)-^ 

71 .  The  6th  term  of  (x^y  +  3  yVx)^. 

72.  The  5th  term  of  (x^  -  3  x-\/y)-\ 

73.  The  7th  term  of \^=.- 

a  —'-wax 

74.  The  8th  term  of  -     ^^ 

Vl-4a26 


148  LOGAmTBMS 

75.  The  5th  term  of  '^{a-Q,^cy. 

76.  The  10th  term  of — 

77.  The  9th  term  of  (V2  -  3 V^^)-\ 

78.  The  8th  term  of  (2  +  V^^)~^. 

79.  The  term  containing  a^  in  (1  -\-x)~^. 

80.  The  term  containing  ic"  in  (x'^  —  2  x)~\ 

81.  The  term  containing  x~^  in  (v^  — 4Va)^. 

82.  The  term  containing  a;"'^^  in  T^^-  ?:^T . 

LOGARITHMS 

Rsercise  62 

Compute  by  four-place  logarithm  tables  the  values  of  the 
following : 

1.  55x3.86.  7.  823x756-- 4320. 

2.  7.81  X  9.3  X  .568.  ^  7.61  x  53.8  x  4.113 

3.  8.52  X  .0917  X  63.4.  *         27.5x1.884 

4.  .097x63.8x51.14.  _       328x57.42 

<7. 


5.  8.76  X  95.32 -- 614.3.  134.2x3.876 

6.  71.3x5.888-43.18.  10.   123.5x3.586-976.3. 

11.   36.95  X  438.7  -  (356.7  X  81.44). 

87.63  X  563.8  x  .0075  91.76 x. 00385x2.1 176 

27.51  X  9832  x  .0953  '  *    7.143  x  .08885  x  11.58  ' 


LOGARITHMS 


149 


14. 


75  X  .03896  x .4427 
83  X  .008114  X  7.003" 


15. 


876.4  X  3.175  x  .6511 
8.465  x  .1973  x  598.6* 


16. 


17.    ■v^^9?f6. 


18.    V17  X  29. 


19.  ^/365.4. 

20.  V.0837. 


29 


5  X  78.3 


X19.7 


30 


31. 


'■i 


9.02  X  1.762 
3.117  X  .0585' 


5/17.44  X  .( 
\zt2  11  V  8 


.0832 
'42.11x8.104* 


7.663  X  85.12  x  .00681 
43.27  X  95.16  x  .007194* 

21.  -v/^00302.  25.  -J/7.13  x  41.2. 

22.  -v/.00075.  26.  V10.3^.049. 

23.  V93  X  2.78.  27.  ^361^5.88. 

24.  ^951  X  .037.  28.  ^8.95 -f- 16.44. 

3^     853.4  X  V:004176 
627.1  X  ■</  .06329  * 

-.     ^5/57.18  X  3.876 
^^'    \7.116x.0 

39 


X  .0485 


7/5.192  X  V63l8 
^  81.32  X  .0638  * 


32 


■41 


382  X  763.5 


.03871  X  8124 


33. 


34, 


35. 


36. 


4 


03765  X  1.448 


37.12  X  5.718 

.4138  X  VaiTr 
3.108  X  i/Tmi 

5.167  X  V38:27 
77.38  X  ^:09034 

.7563  X  -^2:087 
.5432  X  </M15 


40 


41 


3/.07162  X  .00328, 
^1.586  XA/37f77 

^/.0913  X  \/:07652 
'    ^    1.517  X  7.038 


42. 


43. 


V 


3/1.716  X  873.5 
.0173xV3967 


V 


750  x  (.83) 


97  X  (.0361)* 


44 


3/  92.3 


X  .08763 


.003151  X  5V30 


150  LOGARITHMS 

45     J9.3I0  X  -5^7^  4g     37.5  6/440V:0074 

^10.14  xV:3876  *    583^  19^600 

47.  V8.5I6  X  v^9L763  x  \/l998. 

48.  \/53:34xa/7JI6-V98J[5. 

49.  V8.176  X  VMM  X  VTlI^. 

50.  -v^l.Tie  X  8.513  X  VtKM. 
61.  V73.14X  80.37 -^5768. 
52.  ^53.11  X  V7:852  -  3  VTT^. 
63.  •v'3V85  X  2^916  x  5^45. 


g4     J876.3  X  5.173  x  V.()08886 
6.385  X  711.5  X  v':0l776  * 

gg     J.07138  X  V.00885  x  ^1:762 
.08195  X  V.00176  x  VsM2 

gg^    J57I.2  X  (3.817)'  X  (.07161)^^ 
88.19  X  (2.716)^  X  (1.558)* 

Solve,  by  logarithms,  the  following  equations : 

57.  a;  =  19V2l.  61.   38a^  =  235. 

oa  62.    3V^  =  17. 

58.  x  =  -^' 

Vll  63.    11^  =  13. 

59.  a;ViO  =  95.  64.    a;V5  =  -5^'50. 

60.  a^  =  3.47.  65.    97ic2  =  V855. 


LOGARITHMS 

151 

66.  Vl7a;  = 

67.  a/5.5  a^: 

:  9.74. 
=  V79. 

' 

72. 

38^/^     a;V55 
97           79 

68.    30xV7: 

=  6iA/m 

73. 

^^/  =  V190. 

69.    13A/15a 

:  =  27V: 

Oil. 

70.    26Vx  = 

a;V33.7. 

74. 

17  a;"^  =  65. 

71.     ^'''-. 
VSx 

^4l 
So; 

75. 

xVSf     ^38 
75         V^ 

76.    5^  =  30. 

81 

.   65' =  3. 

86.  25*-i  =  ll*. 

77.    6''  =  75. 

82, 

.   40*  =  5. 

87.    552-- =  21*+^ 

78.    7^  =  15. 

83, 

.    18.6' =  1.86. 

88.   (3  +  .05)^^=100. 

79.    8^  =  100. 

84, 

.   9*+i  =  15. 

1 
89.  .9'"^  =  4.7-i 

80.    4.5*  =  50. 

85. 

.   3^-1  =  36. 

90.  (1.025)»«=1.01. 

Calculate : 

91.   Iog4  20. 

95.    Iogi6  60. 

99.    log2jj.68. 

92.   logy  500 

96.    logijS. 

100.   logi2.3 .0423. 

93.    logs  35. 

97.    loggl.O. 

101.   log.5.63. 

94.    logi3  29. 

98.    Iog2o.4. 

102.    log.8.07. 

103.  Find  the  amount  of  $  575  in  10  years  at  5%,  compound 
interest. 

104.  Find  the  amount  of  $  8500  in  12  years  at  4%,  compound 
interest. 

105.  Find  the  amount  of  $3500  in  6  years  at  6%,  compound 
interest. 

106.  At  what  rate  will*  $  12,000  amount  to  f  14,587  in  4  years  ? 


152  LOGARITHMS 

107.  At  what  rate  will  $  8250  amount  to  $  11,627  in  10  years  ? 

108.  What  sum  will  amount  to  $520.75  in  6  years  at  5%, 
compound  interest  ? 

109.  What  sum  will  amount  to  $817.30  in  8  years  at  7%, 
compound  interest  ? 

110.  In  what  time  will  $5000  amount  to  $8000  at  5%, 
compound  interest  ? 

111.  In  what  time  will  $2750  amount  to  $5000  at  4%, 
compound  interest? 

The  following  examples  in  this  exercise  are  to  be  done  with- 
out the  use  of  logarithmic  tables. 

112.  Find  the  logarithm  of  27  to  the  base  3. 

113.  Find  the  logarithm  of  3  to  the  base  27. 

Find : 

114.  log4  8.  118.  logai-  122.  log^^  81. 

115.  log25l25.  119.  logi8.  123.  logaayV 

116.  log27  81.  120.  logioolO.  124.  log64  3V 

117.  logs  32.  121.  log^27.  125.  log^ooo  .01. 

Find  a;,  if : 

126.  log,  8  =  3.  132.  log,  27  =-3. 

127.  log,  81  =  4.  133.  log,  64  =  -  If 

128.  log,  125  =  3.  134.  log,  7  =  -  i 

129.  log,6  =  f  135.  log,  2V  =  - 11 

130.  log,  27  =  IJ.  136.  log,  243  =  -  2.5. 

131.  log,32  =  l|.  137.  log,  J?^  =  -  .75. 


LOGARITHMS  153 

Find  x,{f: 

138.  log,  ic  =  4.  142.  log49a;  =  i  146.  log27a;  =  — IJ. 

139.  logs  a;  =  3.  143.  \og3QX  =  —  ^.  147.  logi  x  =  —  .5. 

140.  log9a;  =  |.  144.  log64a;  =  — If.  148.  logo  a;  =  —  3. 

141.  log8aj  =  |.  145.  logia;  =  1.5.  149.  logjg  a;  =  —  1.5. 

Write  out  as  a  polynomial : 

150.  logaV.  ,^^     ,      11  a;^' 

157.    log——' 

151.  logafe^  ^^y 

152.  log^^l  158.    \og4.(x-yy. 

153.  log^.  ^^^'    log  8 «'(«'- ^')- 

c^  160.    log3a(9-n2)i 

154.  log3aa;^.^  ^^^     log  a  (a  +  5)>3  _  ^s). 

155.  log -2^-  162.    log8a6«-\/5a=^6-^c-i 

156.  logL^!^.  163.    log?^M±^. 

Change  to  the  logarithm  of  a  single  term : 

164.  log  a  4- 3  log  6.  167.    log  11  4- i  log  a. 

165.  log  7  — 3  log ».  168.    31oga-21og6. 

166.  logm+i^gT.  169.    log  2  +  log  3  -  log  13. 

170.  2  log  3  +  3  log  a  —  log  5  —  i  log  x. 

171.  log5  +  31ogaj4-ilog2/— 21og7  — ^loga. 

172.  1  log  2  —  J  log  a;  +  2  log  a  —  I  log  2/  —  log  z. 

173.  log  7  + 1  log  a  4-  f  log  &  -  log  6  -  ^  log  c. 

174.  log  (a  +  1)  +  log  (a  —  1)  —  log  2  —  J  log  m. 

175.  ilog(i)  +  g)+ilogO-g)-ilog(a;  +  ^)-ilog(a; -»!/). 


154  LOGARITHMS 

176.  log  3  +  2  log  2  +  i  log  5  -  log  7  -  log  11  -  f  log  13. 

177.  31og2  +  l+|log7-ilog3-^log(a2  +  62)_iog^^ 


Find  a;,  if: 

178.    a^  =  bc^. 

182. 

3a^-2  =  d^ 

179.    3b'  =  cd'. 

183. 

a6-  =  cd'-\ 

180.    5m'  =  ?t2p2*. 

184. 

m^n''-^  =  p. 

181.    a^-^  =  b\ 

185. 

(^d  =  l^^hn'-\ 

186.  If  log  365  =  2.5623,  write  log  3.65.     Log  .00365. 

187.  If  log  7.008  =  0.8456,  what  is  log  7008  ?     Log  70.08  ? 

188.  If  log  27.8  =  1.4440,  write  log  27800.     Log  .00278. 

189.  If    log  536  =  2.7292,    and    log  537  =  2.7300,    what    is 
log  5.365  ?     What  is  log  .05368  ? 

190.  If    log  3.71  =0.5694,   and    log  37.2  =  1.5705,   what   is 
log  37140  ?     What  is  log  .003717  ? 

191.  If     log  709  =  2.8506,     and    log  7100  =  3.8513,     find 
log  .07096.     Find  log  70.94. 

192.  If  log  627  =  2.7973,  and  log  628  =  2.7980,  find  x,  if 
log  X  =  0.7975.     If  log  X  =  8.7978  -  10.     If  log  x  =  3.7976. 

193.  If  log  3.35  =  0.5250,  and  log  33.6  =  1.5263,  find  x,  if 
log  x  =  9.5254  - 10.     If  log  x  =  7.5260-10.    If  log  x  =  4.5258. 

194.  If  log  2.357  =  0.37236,  log  235.8ii  2.37254,  and    log  x 
=  3.37243,  find  x.     Also  find  x,  if  log  x  =  7.37251  -  10. 

Given,  log  2  =  0.3010 ;  find  the  following  logarithms : 

195.  log  4;  log  40;  log  8;  log  800;  log  32;  log  3.2. 

196.  logV2;  log  ^2;  logV^;  logv'S;  log  Vi02. 

197.  log  5;  log  50;  log  V5;  log  2.5;  log  12.5;  log6J;  log  f . 


LOGARITHMS  155 

Given,  log  2  =  0.3010  and  log  3  =  0.4771 ;  find  the  following 
logarithms : 

198.  log  6;  log  12;  log  18;  log  15;  log  150;  log  14.4. 

199.  log2V3;  log3V2;  logVSO;  log^iOOS;  log  45. 

200.  log  540;  log  .024;  logS^;  log  4^;  log3|. 

201.  From  log  16  how  can  one  get  log  2?    Log  4?    Log  8? 
Log  5?    Log  25? 

202.  From  log  2  and  log  15  how  can  one  find  log3  ? 

203.  From  log  5  and  log  14  how  can  one  find  log  7  ? 

204.  From  log  50  and  log  36  how  can  one  find  log3  ? 

205.  From  log  14,  log  15,  log  16,  how  can  one  find  the  log- 
arithms of  all  numbers  from  1  to  10  ? 

206.  Show  that  there  will  be  31  figures  in  the  100th  power 
of  2.     [log  2  =  0.3010.] 

207.  How  many  digits  in  49^  ?   [log  7  =  0.8451.] 

208.  Find  a;  if  a*' =  &. 

209.  Findajif  a2-'  =  2&.     If2a2-^  =  2*. 

210.  Find  x,  (a)  if  .6^  =  3,  (6)  if  .08^  =  .9,  provided  it  is  given 
that  log  2  ==  0.3010,  log  3  =  0.4771. 

211.  What  is  the  base  if  log  .25  =  - 1  ?     If  log  ^  =  2.5  ? 

212.  Solve  22^-y  =  32  and  3^+^  =  81. 

213.  Find  x  and  y,  if  4^+^^  =  128  and  25^"^  =  125. 

214.  If  log5  =  0.6990,  findxin  the  equations  2*' =  40  and 
(2*)' =  40. 

215.  Show  that  log  |  +  log  f^  -  2  log  .4  =  log 3. 


GENERAL   REVIEW 
Exercise  63 

1.  Solve  the  equation  12a^  —  17  x  =  40  for  x. 

2.  Tell  by  inspection  the  sum,  product,  and  nature  of  the 
roots  of  3a^-lla;  +  15  =  0,  and  of  3a^  +  lla;  =  -8. 

3.  Find  the  sum  of  the  series  4i,  IJ,  |,  •••,  to  infinity. 

4.  Solve  for  x  and  y,  x^  —  jf  =  152,  and  x  =  %  -\-y. 

5.  Define  quadratic  equation,  pure  quadratic,  symmetrical 
expression,  homogeneous  expression,  logarithm,  arithmetical 
progression,  geometrical  progression,  alternation,  composition, 
and  mean  proportional.     Give  an  illustration  of  each. 

6.  Form  that  quadratic  whose  roots  shall  be  1\  and  —  2|-. 

Also  that  one  with  — = for  roots. 

2 

7.  In  the  A.  P.  15,  131,  12,  ••.,  find  the  55th  term  and  the 
sum  of  the  first  20  terms. 

8.  Solve  the  equation  a^  —  1000  =  0  for  its  3  roots. 

9.  Solve  2x^  —  3xy  -{-y^  =  3,  3x^  —  xy  =  2,  ioi  x  and  y. 

10.  Which  term  in  the  series  2|,  3,  3|,  •••,  is  65  ? 

11.  Find  all  the  values  of  ic  in  x^  =  9x. 

12.  When  are  the  roots  of  la^  -\- mx  +  n  =  0  real  ?     When 
equal  ?     When  irrational  ?     When  imaginary  ? 

13.  From  the  letters  in  the  word  scholar,  how  many  words 
can  be  formed,  of  4  letters  each  ?     Of  7  letters  each  ? 

166 


Gi:Ni:iiAL  review  157 

1  _3 

14.  Solve 7  X  ^  =  8.     Are  both  the  values  of  x  roots  ? 

x^ 

15.  Find  the  values  of  a?  and  2/  in  ar^— 2a;2/— a;=3,  3ic— 42/=7. 

16.  Find  the  values  of  x,  by  inspection,  in 

2aa;(ar'-4)(3a^  +  5)  =  0. 

17.  What  is  the  logarithm  of  216  to  the  base  36?  Of  8  to 
the  base  128  ?     What  is  \og„  ^J^  ? 

18.  It  a:b  =  c:  d,  prove  that 

19.  Write  the  equation  32^  ^  =  64  in  logarithmic  form.  What 
is  the  number?  The  logarithm?  The  base?  The  charac- 
teristic?   The  mantissa?    Write  logs;  243  =  1|  as  an  equation. 

20.  How  many  parties  of  7  each  can  be  selected  from  a 
school  of  25  girls  ? 

21.  Solve  2-\/x  +  3  —  Va;  —  2  =  V3  x  —  2.  Are  both  values 
of  X  roots  of  this  equation  ?     Why  ? 

22.  Find  the  6th  term  and  the  sum  of  7  terms  in  the  series 
-4,6,-9,.... 

23.  Solve  1^=^-2 -^^  =  0. 

S-x  x+2 

24.  Prove  the  formula  for  sum  of  a  G.  P.  if  the  first  term, 
last  term,  and  ratio  are  given.  Derive  formula  for  sum  of 
infinite  geometrical  series. 

25.  Solve  a (6  a  —  13)  —  2  ax (n  —  x)  =  5(x-\-  3). 

26.  Why  cannot  a  negative  number  have  any  logarithm  pro- 
vided the  base  is  always  positive  ?  How  are  operations  with 
negative  numbers  performed  by  aid  of  logarithms  ? 

27.  Expand  (2-Vx  -  x^^/yy. 


158  GENERAL  REVIEW 

28.  Find  the  values  of  a  that  will  make  the  equation 
4:X'  —  lox—ax-^a-{-20  =  0  have  equal  roots.  Prove  your 
values  correct. 

29.  In  an  A.  P.,  Z  =  14,  w  =  40,  s  =  430 ;  find  a  and  d 


30.  Solve  2Va;-2  +  V2a;  +  3  =  V8a;  +  l. 

31.  From  the  letters  in  the  word  sweetest,  how  many  arrange- 
ments can  be  made,  taking  all  the  letters  every  time  ? 

32.  Find  the  middle  term  of  (a^y —  ^Vx~'^y^. 

33.  Solve  a^ -{- xy  —  y^  =  1,  xy-\-2y^  =  3. 

34.  Insert  5  geometrical  means  between  2|  and  30f. 

35.  Prove    the  formulas  for  /  in  the  progressions.      Also 
prove  the  formulas  for  sum  in  arithmetical  progression. 

36.  Solve  for  x,  2  V3a;-2  -  3  ^3x^-2  =  2. 


37.  Compute  by  logarithms,    Y^^'^  ^  ^-^^^- 

</.0716  X  438.6 

38.  Form  the  quadratic  whose  roots  will  be  —a  and  ^-^ — 
The  quadratic  whose  roots  are  -^ — ^^— — 

39.  Distinguish  between  mean  proportional  and  third  pro- 
portional. Find  the  mean  and  the  third  proportional  to  a 
and  2ab. 

40.  From  a  class  of  8  boys  and  10  girls,  how  many  groups 
of  2  boys  and  3  girls  each  can  be  selected  ? 

41.  Solve  2^^ -3  =  -^  +  ^. 

3x  +  4:  6a;H-3 

42.  Insert  23  arithmetical  means  between  3  and  63.  Also 
35  means  between  ^  and  27.5. 

43.  Solve  x-\-2y  =  2,  a^  +  Sf  =26. 


GENERAL  REVIEW  159 

44.  Solve  (x^  +  x-5y-S(x^-\-x) +4:7  =  0. 

45.  From  16  consonants  and  4  vowels,  how  many  words  can 
be  formed,  each  consisting  of  3  consonants  and  2  vowels  ? 

46.  Compute  by  logarithms  the  value  of  J-8034  x  VTOgSTg^ 

.5138  X  ^.00175 

47.  A  man  agreed  to  dig  a  well  at  the  rate  of  25  cents  for 
the  first  yard,  50  cents  for  the  second,  7.5  cents  for  the  third, 
and  so  on.  Upon  completion  he  received  $30.  How  deep 
was  the  well  ? 

48.  Find  the  value,  by  logarithms,  of  2^  x  (i)^  X^/^X  VJ. 

49.  What  is  the  value  of  (x  -  V2y  +  (a;  +  V2)*  ? 

50.  How  many  figures  in  Q5^  ? 

51.  Solve  a^  — a;?/  +  22/  =  4,  2x-\-3y  =  5a-}-l. 

52.  li  a:b  =  c:  d,  prove  a^  -f  4  6- :  c^  +  4  c?^  =  a6  :  cc?. 

53.  Prove  the  formulas  for  the  number  of  permutations  and 
the  number  of  combinations  of  m  things  taken  ?i  at  a  time. 

54.  Form  the  quadratic  whose  roots  are  0,  — — .     Also  the 

one  whose  roots  will  be  ^ — — — — 

7 

55.  Solve  (2  a; -3)2 -(a; -1)2  =  5. 

56.  Express  as  a  polynomial,  log  ^  a^^/b ;  log  J x^s/S y. 

57.  Compute  by  logarithms,  — ■T\\h7;:7:' 

^         -^      °     .         '  101  ^'.SOO 


58.  Simplify   {-y/l-  o^  J^lf  -  {■y/l-x'  -If, 

59.  How  many  combinations,  each  containing  6  white  balls 
and  5  red  balls,  can  be  selected  from  14  white  and  10  red  balls  ? 

60.  Solve  2a^  +  3a;2/-52/2  =  4,  2a;?/  +  3/  =  -3. 


160  GENERAL   REVIEW 

61.  Find  x  that  will  make  2x  —  l,  x-^7,  3a;-f-l,  in  A.  P. 
Find  X  if  they  are  in  G.  P. 

62 .  Find  s  that  will  m  ake  the  roots  of  3  sor^ + 2  sa; + 9  a; + 8  =  0 
equal  to  each  other. 

63.  li  a  :  b  =  c  :  d,  prove 


Va=*  H-  b' :  Vc'  +  d'  =  i/a'  -  b^ :  ^/(^-(j^. 

64.  Solve  8a.--3  +  llVaP  =  54. 

65.  Twenty  men  are  going  to  march  four  abreast.  In  how 
many  ways  can  they  place  themselves  ? 

66.  Expand  (V2a-Va;)®. 

67.  Solve  y-l  =  x--  =  -' 

X  y     2 

68.  Solve  2  a^-aa;-a2  =2  (3  a; -2),  for  ir. 

69.  Find,  by  logarithms,  the  value  of  x,  if  7^  =  100. 

70.  Without  solving  the  equations,  tell  what  is  the  sum, 
product,  and  character  of  their  roots :  (a)  3a^  —  7a;  =  —  2; 
(6)  5ar'  +  4a;  +  l  =  0. 

71.  What  is  the  value  of  2.7181818  -.  ? 


72.  Solve  2a^-5a;-V2a^-5a;  +  l  =  l. 

73.  Form  that  quadratic  the  sum  of  whose  roots  is  11  and 
whose  product  is  13^.     Prove  your  answer. 

74.  If  a:b  =  b:c=c:d,  prove  that  a-{-b:b-{-c=2  a  —  b:2  b—c. 

75.  ^o\Yeabx'-a\x-l)  =  b\x-\-l). 

76.  Find  all  the  values  of  x  and  y  in  the  equations  (x-\-yy 
-x-y  =  Q>,  x'y'^  +  lxy +  12  =  0. 

77.  The  sum  of  35  terms  of  an  A.  P.  is  490  and  the  com- 
mon difference  is  J.     Find  the  first  and  last  terms. 


GENERAL  REVIEW  161 

78.  If  m  and  n  are  the  roots  of  ic^  — 07  +  1  =  0,  show  that 

m^-{-n^ +  1  =  0.     Show  that  i  +  ^  =  l.     That  m-n  =  V^^. 

m     n 

79.  The  sum  of  five  terms  of  a  G.  P.  is  5|^  and  the  ratio  is 
|.     Find  the  first  and  last  terms. 

80.  Find  the  four  roots  of  8  a;^  =  27  x. 

81.  Prove  that  the  roots  of  ax^ -\- 2  bx -{- c  =  0  will  be  equal 
if  6  is  a  mean  proportional  between  a  and  c.  When  will  the 
roots  be  rational  ? 

82.  Expand  (1  —  4  x)~^  to  5  terms. 

83.  Find  the  limit  of  the  sum  of  1  —  |  +  ^.•.  to  infinity. 


84.  Solve4cc  +  4V3a^-7a;  +  3  =  3ar^-3a;  +  6. 

85.  Find  two  numbers  whose  difference,  sum,  and  product 
are  to  each  other  as  2:3:5. 

86.  There  are  three  numbers  in  A.  P.  whose  sum  is  3.  If 
3,  4,  and  21  be  added  to  them,  respectively,  the  sums  form  a 
G.  P.     Find  the  numbers. 

87.  How  many  terms  of  the  series  32,  48,  72,  •..,  amount  to 
665? 

88.  Form  the  equation  whose  roots  are  double  the  roots  of 
x'-3x-\-2^  =  0. 

89.  The  5th  term  of  an  A.  P.  is  —  3  and  the  15th  term  is  17. 
Find  the  sum  of  the  first  20  terms. 

90.  How  many  arrangements  can  be  made  from  the  letters 
in  the  word  holiday,  taken  all  together  ?  How  many,  if  the 
three  letters  lid  are  never  separated  ? 

91.  If  the  base  of  a  system  of  logarithms  be  4,  tell  the  loga- 
rithm of  each:    16;   8;   32;   2;   J;   1;   i;   |;   ^;    Vj;    </l', 


R.   &  S.   EX.  IN  ALG. 11 


162  GENERAL  REVIEW 

92.  Solve  i  +  i  =  l,  1+1  =  13. 

X      y  0^     y^ 

93.  How  many  terms  of  the  series  41  4,  3|,  •••,  amount  to 
21? 

94.  Find  that  G.  P.  the  sum  of  an  infinite  number  of  whose 
terms  is  4  and  the  second  term  is  \. 

95.  Compute    the    5th    root    of      4.281  x  V.09m 
logarithms.  321.7  x  •v/.008074       ^ 

96.  Find  the  10th  term  of  z^ by  the  binomial 

theorem.  (l-2Va^)^ 


97.  Find  all  values  of  a;  in  1+2  a^+3  x=3V2  a^+3  x—\. 

98.  If  a,  6,  c,  dy  are  in  continued  proportion,  prove  that 
lo?  +  mV  +  ne  :  Ih''  -\-m&^  nd?  =  ac?  +  &c  :  2  cd. 

99.  Solve  this  equation  for  the  value  of  a;:  a;  +  -  =  -  +  -. 

a     X     h 

100.  Find  a:  and  ?/  in  a^  +  /  =  3^,  x-^  +  2/"^  =  If 

101.  Find  the  sum  of  all  numbers  between  10  and  500 
exactly  divisible  by  7. 

102.  What  is  meant  by  "  completing  the  square  "  ?  How  is 
it  done  ?  What  is  an  imaginary  ?  What  are  conjugate  im- 
aginaries  ?     Prove  that  log  PQ  =  log  P  +  log  Q. 

103.  From  the  usual  formulas  for  I  and  s  of  an  A.  P.  derive 
a  formula  for  a  not  containing  s.  A  formula  for  I  not  contain- 
ing a.     A  formula  for  d  not  containing  I. 

104.  Out  of  15  consonants  and  5  vowels  how  many  words 
can  be  made  each  consisting  of  4  consonants  and  3  vowels  ? 

105.  Find,  to  3  decimal  places,  the  logarithm  of  65  in  a 
system  whose  base  is  15. 


GENERAL  REVIEW  163 


106.  Solve  Vx  +  y  =  Vy-\-2,  x  —  y  =  7. 

107.  When  are  3  quantities  in  continued  proportion  ? 
Prove  that  if  a,  h,  c,  are  in  continued  proportion,  then  a,  a  +  b^ 
a  +  2  6  +  c,  are  also  in  continued  proportion. 

108.  Find  a;  if  (a^  +  2  xf  -lSx(x  +  2) +45  =  0. 

109.  If  the  base  of  a  system  of  logarithms  is  a,  what  is  log  a? 
log-?  loga^?  logVa?  log  A/a*?  logAZ-i? 

X     y        ^  a^     y^ 


110.    Solve  ±-i:  =  2,  -,--,  =  3i 


111.  Calculate  by  logarithms  the  mean  proportional  between 
V5:082  and  a/.009116. 

112.  In  a  G.  P.  the  5th  term  is  12  and  the  11th  term  is  768. 
Find  the  3d  term  and  sum  of  9  terms. 

113.  Solve =l-i  +  i. 

a-^b  —  x     a     X     b 

114.  Find  a;  if  2.5' =  75. 

115.  How  many  numbers,  of  6  different  digits  each,  can  be 
written  from  our  9  significant  digits  ? 

116.  Find  x  and  y  if  a^  —  a;?/  =  1^-  and  xy-{-y^  =  1. 

117.  Insert  5  geometrical  means  between  5  and  3645.     Also 
69  arithmetical  means  between  5  and  3645. 

, ,  o     n  ^    ^1,        1        f  /  -  4.116  X  75.38  X  .0567  \^ 

118.  Compute  the  value  of  (  — — ,     ^  ^^^.   — 7:7^1^    ' 

^  \S1.24:  X  (  - 1.909)  X  .0053; 

11-9.    Solve  for  x  and  y :  x^-{-xy  +  2/^  =  13 ;  a^  +  a^/  +  2/^  =  91. 

120.  Find  x  and  y  in  x^ +  y^-{-x-\-y  =  26;  ccy  =  — 10. 

,«-.     ai      J?  a.'  — 1      -,      3a!-|-2      ^ 

121.  Solve  for  x : —  1 -!— -  =  0. 

a;  —  3  a;  +  3 


164  GENERAL  REVIEW 

122.  Given  log  40  =  1.60206,  find  log  2;  log  5;  log  20;  log 

50;  log  V5;  log  ■y/2.5',  log  i;  log  ^ ;  log  |;  log  1.25;  log  2i. 

123.  li  S'.t  =  u'.v  =  w:x  =  y:Zf  prove  that 

s-\-u-\-w-\-y'.t-\-v-\-x-\-z  =  s:t  —  etc. 

124.  Find  all*  the  values  of  a;  in :  3  a^x  (x  +  1)  (a;^  -  81)  =  0. 

125.  Solve  for  x  and  y:  x^  -\-  y'^  -\-  x  —  y  =  2Q  -^  xy  -\-16  =  0. 

126.  From  a  delegation  of  15  Protestants  and  11  Catholics, 
there  is  to  be  chosen  a  committee  of  6  Protestants  and  4  Catho- 
lics.    In  how  many  ways  might  this  be  accomplished  ? 

127.  lix^  -\-y^  =  ^  and  ic  +  ?/  =  41,  find  x  and  y. 


128.  If  V3a;-2  +  VaJ  +  2  -4  =  0,  find  x  and  discuss  its 
values  as  roots. 

129.  Given  a  =  2,  Z  =  ^,  s  =  \2^,  find  r  and  w. 

130.  Expand  (2  -  V^^)^ 

131.  Solve  cc^  —  2/^  =  7,  a;  —  2/ =  133. 

132.  The  5th  term  of  a  G.  P.  is  336  and  the  9th  term  is 
5376.     What  is  the  2d  term  ? 

133.  Finda::^±^^^^  =  :^^^+1.     Discuss  its  values. 

X  —  V12  a  —  x      -y/a  —  l 

134.  If  a  certain  number  is  divided  by  8,  the  result  will  be 
the  same  as  if  16  were  divided  by  the  number  and  3J  added 
to  the  quotient.     What  is  the  number  ? 

135.  If  V  :x  =  x:y  =  y  :z,  prove  that  x-^-y  is  a  mean  pro- 
portional between  v-\-x  and  y  -\-z. 

136.  If  aj  +  2/4-l  =  a^-/  +  a:2/-l  =  0,finda;andy. 


GENERAL  BEVIEW  165 

137.  What    are    the    values    of     x     in     the     equation 
^    —       ~    =4?    Are  these  values  both  roots? 

138.  Solve  x^  —  2ax  —  2bx  +  (a  +  b-\-c)(a-\-b  —  c)  =  0. 

139.  The  sum  of  the  first  two  terms  of  a  G.  P.  is  72  and  the 
sum  of  the  next  two  is  8.  What  is  the  1st  term  ?  The  5th 
term  ? 

140.  Expand  to  4  terms :  (y~^  —  5  a?^  —1)'^. 

141.  Find  the  series  in  which  d  =  8,  1  =  147,  and  s  =  1425. 

142.  If  the  ratios  I :  m,  n  :  p,  q :  r,  are  equal,  prove  that  each 


is  equal  to  J ''+/+/, 


m^ -{- p'^ -{- 7^ 

143.  Solve  (3  a;  -  2)2  4- («  - 1)  =  84. 

144.  Prove  the  binomial  formula  for  positive  integral 
exponents. 

145.  What  is  the  7th  term  of  (x-^  -  ^Vxf)'^  ? 

146.  ^olYex^-\-xy-^2y^  =  ll  =  a^-\-§y\ 

147.  Find  a;  if  2^' =  500. 

148.  From  the  figures  1,  2,  3,  4,  5,  6,  7,  how  many  numbers 
can  be  formed  of  5  different  figures  each  ?  How  many  of  these 
will  contain  a  3  ?  How  many  will  have  3  and  6  together  ? 
How  many  will  be  odd  ? 

149.  Multiply  .03716^  by  1.8716^  by  logarithms. 

150.  If,  in  an  A.  P.,  a  =  s  =  —  |  and  n  =  20,  find  d  and  L 

151.  Find  that  G.  P.  whose  sum  to  infinity  is  1^  and  whose 
2d  term  is  ^. 


152.    Solve  -^ fl  +  J_^a;  +  i  +  l  =  0. 

m  -{-  n      \       mnj        m      n 


166  GENERAL  REVIEW 

153.  Solve  -  +  i  =  -^,   a;v  =  54. 

ic      2/      18 

154.  Insert  6  geometrical  means  between  5  and  —640. 

155.  Find  x  and  y  if  3*  5^  =  75,  2*  7^  =  98. 

156.  How  many  different  sounds  can  be  made  by  striking 
16  keys  of  a  piano,  3  at  a  time  ? 

157.  If  2(V^-3)2-3  =  V^,  find  x, 

158.  Expand  (2V3+3V2)^ 

159.  Find  x  and  y  from  x  +  y  =  ^,  x^y^  —  2^xy  =  — 192. 

160.  Solve  6a^-3a;  =  2  +  V2a^-a;. 

161.  There  are  two  fractions  the  sum  of  whose  denomina- 
tors is  5.  The  numerator  of  the  first  is  the  square  of  the  de- 
nominator of  the  second,  and  the  numerator  of  the  second  is 
the  square  of  the  denominator  of  the  first.  The  sum  of  the 
fractions  is  5|.     Find  them. 

162.  The  sum  of  an  infinite  number  of  terms  of  a  certain 
G.  P.  is  4i  The  sum  of  the  first  two  terms  is  2|.  Find 
the  series. 

163.  If  log  13  =  1.1139,  what  is  log  a/1300  ?     log  V.0013? 

164.  Find  a;:   Va^-fl:  \/2x  =  ^       ^ 


x    a;-f 4 
165.   Evaluate  the  decimal  1.4363636—. 


166.  Solve  9a;-3x2-4Vaj2-3a;-f  5  =  0. 

167.  Out  of  7  consonants  and  4  vowels  how  many  words 
can  be  formed,  each  consisting  of  3  consonants  and  2  vowels  ? 

168.  Solve  for  x :  (x-^  +  i)"'  =  27  and  (a?"?  ^  |)-i  =  -  \. 


GENERAL  BEVIEW  167 

169.  Find  the  numerical  value  of 

i  logs  9-2  log27  3  +  log,  a?  -  log,  1. 

170.  Show  that  ma-{-nb:  pa-\-qb=7nG+nd:  pc-{-qd  provided 
a,  b,  c,  d,  are  proportional. 

171.  Find  a;  if  4*^-1  =  (!)*-«.     [Without  tables.] 

172.  How  many  games  must  be  played  in  a  league  of  10 
baseball  clubs,  provided  each  club  plays  10  games  with  every 
other  club  ? 

173.  Insert  5  arithmetical  means  between  —  7  and  77. 

174.  Find  the  values  of  x,  correct  to  two  decimal  places,  in 
the  equation  ic^  —  2  a;  —  2  =  0. 

175.  Solve  iJc^  +  5xy  +  Sf  =  S,  Sx^-^7 xy-{-4:y^  =  5. 

176.  If  the  series  12,  9,  •••,  is  arithmetical,  find  the  sum  of 
20  terms.  If  geometrical,  find  the  sum  of  an  infinite  number 
of  terms. 

177.  Solve  a;^  +  a;^  + 1  =  0. 

178.  Tell  sum,  product,  and  nature  of  roots  of  3  a^  —  4  a;  — 11 

=  0.     Also  of  2a^  +  3ajH-7  =  0. 

179.  Find  the  sum  of  all  the  odd  numbers  between  20  and 
220. 

180.  Expand  (J V3  +  W^-^Y  and  simplify. 

181.  If  a,  b,  c,  d,  are  proportional,  show  that  ab  +  cd  is  a 
mean  proportional  between  a^  +  c^  and  b^  +  d\ 

182.  Solve  the  equations  — 1--= — ,    a^+y^=a\ 

X     y     xy 

183.  Find  two  numbers  in  ratio  of  7:5,  the  difference 
between  whose  squares  is  96. 


168  GENERAL.  BEVIEW 

3/ = 

184.    Find  the  8th  term  of  the  expansion  of  va  — 6Vaa;,  by 
the  binomial  theorem. 


185.    Solvefor?/:  l+V8^-32/2  =  22/. 


186.    Solve  a;  4-  2/  +  2  Va;  +  2/  =  24,  «  -  ?/  +  3  Va;  —  2/  =  10. 


187.  Find  the  value  oiz:  o?z-2h^=  ah 

188.  Solve?  +  -  =  5,   ^,  +  -^-^  =  -19. 

X     y  x-      xy      y^ 


Hint:   Let-  =  w, -  =  w.1 
L  X  y         A 


189.  Discuss  the  values  of  x  in  the  equation, 

V3a;  +  1  =  V9aj  +  4  -  V2  a;  -  1. 

190.  Find,  by  logarithms,  the  values  of  x  and  y^  if  3"=  =  50  ^ 
and  2^  =  b  y. 

191.  Find  the  (r  +  l)th  term  of  (1  -  x)^. 


192.  Simplify  (2  -  Vl  -  xy  +  (2  +  Vl  -  x)\ 

193.  Compute,  by  logarithms,  the  value  of 


^moiW^- 


194.    Solve  Va  —  a;  +  V— (a^  +  ax)  =  -^= 


X 


195.   If  a,  6,  and  c  are  in  G.  P.,  show  that  ,  -,  and 


aremA.P.  «  +  *2'  6  +  <= 

196.   Find  the  3  cube  roots  of  unity ;  i.e.  solve  the  equa- 
tion x^  =  1. 


GENERAL  BEVIEW  169 

197.    Solve  for  x:-\ h  -  ^  = ^^, — 7^ — 

2  I  1  — a      x\  x(a-  —  1) 


198.  Solve   V5a;  +  l  +  2Vaj-2  =  3V2(a;-l). 

199.  Solve   (m  +  xy  —  (m  —  xy  =  x. 

200.  Of  how  many  terms  does  an  A.  P.  consist  in  which 
d  =  'S,  1  =  302,  a  =  5? 


201.  The  sum  of  an  infinite  number  of  terms  of  a  G-.  P.  is  15 
and  the  second  term  is  3^.     Find  the  fourth  term. 

202.  Solve   ^-^  =  ^. 

a^  - 1      x'  +  l     20 

203.  Expand   (V2«— ^3»)*. 

204.  The  difference  between  two  numbers  is  32  and  the 
arithmetical  mean  exceeds  the  geometrical  mean  by  4.  Find 
the  numbers. 

205.  By  the  principles  of  proportion  find  x  if 


Va;  +  2  -  Va;  -  3  ^  Vr)  .c  + 1  -  4  V.t  -  6 
■y/x  +  2  +  ^x^^      V5a.-  +  l  +  4Va;-6 

206.  Find  sum  of  i  —  1  —  f  •••  to  29  terms. 

207.  Find  sum  of  1  + J +  !•••  to  6  terms. 

208.  Solve   aV  —  oi^  —  d^x  —  5ax  =  6a^. 

209.  The  sum  of  10  terms  of  an  A.  P.  is  100  and  the  sixth 
term  is  11.     Find  the  second  term. 

210.  Solve   a^_a;  +  l  +  -_-l =^. 

ar  — a^  +  1      3 


170  GENERAL   REVIEW 

211.  The  sides  of  a  right  triangle  are  in  A.  P.     Prove  that 
they  are  proportional  to  3,  4,  5. 

212.  Find  a;  if 

213.  How  many  terms  of  the  series  IJ,  1,  |,  •••,  must  be 
taken,  that  the  sum  may  be  zero  ? 

214.  Solve  2a^-3x  +  4  =  -— -4 ^• 

2x'-Sx  +  2 

215.  In  an  A.  P.  a  =  7,  d  =  —  i  s  =  55.     Find  n  and  Z. 


216.  If     V<x  +  a; :  Va  +  Va  +  x  =  Va  —  a; :  Va  —  Va  —  x, 
find  a;. 

Solve  the  following  simultaneous  quadratic  equations  and 
associate  the  corresponding  values  of  x  and  y: 

217.  y^  +  xy-^y  =  -6',  x'-\-xy-^x  =  S, 

218.  x(x  -h  2/)  =  10 ;  2/(2/  -  x)  =  3. 


219.  VaH^— Va;  — 2/  =  2;  3aj-22/  =  7. 

220.  3,-3-1  =  316;  a;-i=4. 

r  y 

221.  a^  +  2 a^  =  -  21 ;  y^  ■^2xy=-b. 

222.  «2  +  2/2_4aVa^  4- 2/^  =  5^2;  y?  -  f  :=!  a\ 

223.  a(a;  —  a)  =  6(2/  —  6)  ;  ax  -{-  by  =  xy. 

224.  6a^y^  +  5xy  =  6;  x -\- SO y  =  12. 

o«,  i      (^-32/)^-8a;  +  24^  =  -12; 

''^^'  ^2(2a;  +  2/)2-22a;-lli/  =  ^5. 


GENERAL  REVIEW  171 


226.  x-y-^x  —  y  =  2;  a?-f  =  20^. 

227.  2:f?-xy  =  \2',  a? -2  xy +  Zy''  =  ^. 

228.  ^x^-{-6xy  —  3x  —  y  +  2  =  0'^  2x-\-5y=  —  4:. 


229.  One  root  of  the  equation  a^  — 6ic+29=0is  3— 2V^^. 
Without  solving  this  quadratic,  find  the  other  root  and  prove 
your  answer  correct. 

230.  Solve:  ^- ^=2£l^. 

a;— 1     aj-f-l     a^  —  2a 

231.  Find  the  limit  of  the  sum  of  the  series  5  —  3  +  1  —  ^ 
+  ^  —  •  •  •  to  infinity. 


232.  Solve:  V3a^-2a;  +  4  =  3  0.-2-23;- 8. 

233.  If  w,  X,  y,  and  z  are  proportional,  show  that  -\/w^  —  y^y 
Vor  —  z^,  w-\-  y,  X  +  z,  are  also  proportional. 

234.  Find  the  ninth  term  of  (ia^y-2  ^/xy\ 

235.  Divide  9 J  into  three  parts  in  G.  P.  such  that  the  sum 
of  the  first  two  is  to  the  sum  of  the  last  two  as  3  to  2. 

236.  For  what  values  of  m  will  the  equation  4  ic^  — 15  ic  +  36 
=  m(x  —  1)  have  equal  roots  ?    Verify  your  results. 


172 


LOGARITHMS 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 
12 
13 
14 

0000 
0414 
0792 
1139 
1461 

0043 
0453 
0828 
1173 
1492 

0086 
0492 
0864 
1206 
1523 

0128 
0531 
0899 
1239 
1553 

0170 
0569 
0934 
1271 
1584 

0212 
0607 
0969 
1303 
1614 

0263 
0645 
1004 
1336 
1644 

0294 
0682 
1038 
1367 
1673 

0334 
0719 
1072 
1399 
1703 

0374 
0766 
1106 
1430 
1732 

15 

16 
17 
18 
19 

1761 
2041 
2304 
2553 

2788 

1790 
2068 
2330 
2577 
2810 

1818 
2095 
2355 
2601 
2833 

1847 
2122 
2380 
2625 
2856 

1875 
2148 
2405 
2648 
2878 

1903 
2175 
2430 
2672 
2900 

1931 
2201 
2465 
2696 
2923 

1969 
2227 
2480 
2718 
2945 

1987 
2263 
2604 
2742 
2967 

2014 
2279 
2629 
2766 
2989 

20 

21 
22 
23 
24 

3010 
3222 
3424 
3617 
3802 

3032 
3243 
3444 
3636 
3820 

3054 
3263 
3464 
8655 
3838 

3075 
3284 
3483 
3674 
3856 

3096 
3304 
3502 
3692 
3874 

3118 
3324 
3522 
3711 
3892 

3139 
3345 
3541 
3729 
3909 

3160 
3365 
3560 
3747 
3927 

3181 
3385 
3579 
3766 
3945 

3201 
3404 
3698 
3784 
3962 

25 

26 
27 
28 
29 

3979 
4150 
4314 
4472 
4624 

3997 
4166 
4330 
4487 
4639 

4014 
4183 
4346 
4502 
4654 

4031 
4200 
4362 
4518 
4669 

4048 
4216 
4378 
4533 
4683 

4065 
4232 
4393 
4648 
4698 

4082 
4249 
4409 
4664 
4713 

4099 
4265 
4425 
4579 
4728 

4116 
4281 
4440 
4594 
4742 

4133 
4298 
4456 
4609 
4767 

30 

31 
32 
33 
34 

4771 
4914 
5051 
5185 
5315 

4786 
4928 
5065 
5198 
5328 

4800 
4942 
5079 
5211 
5340 

4814 
4955 
5092 
5224 
5353 

4829 
4969 
6105 
5237 
5366 

4843 
4983 
5119 
5250 
5378 

4857 
4997 
6132 
5263 
6391 

4871 
5011 
6145 
5276 
6403 

4886 
6024 
6169 
6289 
6416 

4900 
5038 
5172 
5302 
6428 

35 

36 
37 
38 
39 

5441 
5563 
5682 
5798 
5911 

5453 
5575 
5694 
5809 
5922 

5465 
5587 
5705 
5821 
5933 

5478 
5599 
5717 
5832 
5944 

5490 
5611 
5729 
5843 
5955 

5502 
5623 
6740 
6865 
6966 

5614 
5635 
6752 
5866 

6977 

5527 
5647 
6763 

5877 
6988 

6539 
5658 
6776 
6888 
6999 

6551 
5670 
6786 
6899 
6010 

40 

41 
42 
43 
44 

6021 
6128 
6232 
6335 
6435 

6031 
6138 
6243 
6345 
6444 

6042 
6149 
6253 
6355 
6454 

6053 
6160 
6263 
6365 
6464 

6064 
6170 
6274 
6375 
6474 

6076 
6180 
6284 
6385 
6484 

6086 
6191 
6294 
6395 
6493 

6096 
6201 
6304 
6406 
6503 

6107 
6212 
6314 
6415 
6513 

6117 
6222 
6325 
6425 
6522 

45 

46 
47 
48 
49 

6532 
6628 
6721 
6812 
6902 

6542 
6637 
6730 
6821 
6911 

6551 
6646 
6739 
6830 
6920 

6561 
6656 
6749 
6839 
6928 

6571 
6665 
6758 
6848 
6937 

6580 
6676 
6767 
6857 
6946 

(5590 
6684 
6776 
6866 
6955 

6599 
6693 
6785 
6876 
6964 

6609 
6702 
6794 
6884 
6972 

6618 
6712 
6803 
6893 
6981 

50 
51 
52 
53 

54 

6990 
7076 
7160 
7243 
7324 

6998 
7084 
7168 
7251 
7332 

7007 
7093 
7177 
7259 
7340 

7016 
7101 
7185 
7267 
7348 

7024 
7110 
7193 

7275 
7366 

7033 
7118 
7202 
7284 
7364 

7042 
7126 
7210 
7292 
7372 

7050 
7135 
7218 
7300 
7380 

7059 
7143 
7226 
7308 
7388 

7067 
7152 
7235 
7316 
7396 

LOGARITHMS 


173 


No. 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

55 

56 
67 
58 
59 

7404 

7482 
7559 
7634 
7709 

7412 
7490 
7566 
7642 
7716 

7419 
7497 
7574 
7649 
7723 

7427 
7505 
7582 
7657 
7731 

7435 
7513 
7589 
7664 
7738 

7443 
7520 

7597 
7672 
7745 

7451 

7528 
7604 
7679 

7752 

7459 
7536 
7612 
7686 
7760 

7466 
7543 
7619 
7694 

7767 

7474 
7551 

7627 
7701 

7774 

60 

61 
62 
63 
64 

7782 
7853 
7924 
7993 
8062 

7789 
7860 
7931 
8000 
8069 

7796 
7868 
7938 
8007 
8075 

7803 
7875 
7945 
8014 
8082 

7810 
7882 
7952 
8021 
8089 

7818 
7889 
7959 
8028 
8096 

7825 
7896 
7966 
8035 
8102 

7832 
7903 
7973 
8041 
8109 

7839 
7910 
7980 
8048 
8116 

7846 
7917 
7987 
8055 
8122 

65 

66 
67 
68 
69 

8129 
8195 
8261 
8325 
8388 

8136 
8202 
82(57 
8331 
8395 

8142 
8209 
8274 
8338 
8401 

8149 
8215 
8280 
8344 
»407 

8156 
8222 
8287 
8351 
8414 

8162 
8228 
8293 
8357 
8420 

8109 
8235 
8299 
8363 
8426 

8176 
8241 
8306 
8370 
8432 

8182 
8248 
8312 
8376 
8439 

8189 
8254 
8319 
8382 
8445 

70 
71 
72 
73 
74 

8451 
8513 
8573 
8633 

8692 

8457 
8519 
8579 
8639 
8698 

8463 
8525 
8585 
8645 
8704 

8470 
8531 
8591 
8651 
8710 

8476 
8537 
8597 
8657 
8716 

8482 
8543 
8603 
8663 
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Phillips  and  Fisher's  Geometry 


By  ANDREW  W.   PHILLIPS,   Ph.D. 

and    IRVING    FISHER,   Ph.D. 

Yale   University 

PHILLIPS  AND  FISHER'S  ELEMENTS  OF  PLANE  AND  SOLID 

GEOMETRY $1.75 

PHILLIPS  AND   FISHER'S   PLANE   AND   SOLID   GEOMETRY. 

Abridged $1.25 

PHILLIPS  AND  FISHER'S  PLANE  GEOMETRY— Separate  .  80  cents 
PHILLIPS  AND  FISHER'S  GEOMETRY  OF  SPACE— Separate  $1.25 
PHILLIPS  AND  FISHER'S  LOGARITHMS  OF  NUMBERS  .    30  cents 

The  publication  of  this  text-book  marks  a  new  era  in  the  teaching 
of  Geometry.  Its  distinctive  qualities  are  :  (i)  clearness  of  presentation, 
both  in  form  of  statement  and  in  the  diagrams  ;  (2)  natural  and  sym- 
metrical methods  of  proof ;  (3)  abundance  and  variety  of  original 
problems  for  demonstration  and  for  numerical  computation. 

But  the  feature  which  more  than  any  other  distinguishes  it  from 
similar  text-books  is  the  use  of  photo-engravings  of  geometrical  figures 
arranged  side  by  side  with  skeleton  drawings  of  the  same,  whereby  the 
most  magnificent  collection  of  geometrical  models  ever  constructed  is 
brought  within  reach  of  every  preparatory  school  and  college  student. 
By  this  method  of  illustration  the  great  problem  of  educating  the 
student's  imagination  to  a  proper  comprehension  of  the  figures  of  solid 
geometry  is  practically  solved. 

The  Abridged  Edition  is  intended  for  those  schools  which  desire  a 
briefer  course  than  that  offered  in  the  complete  work.  It  has  all  the 
excellencies  and  features  of  the  larger  book,  including  the  reproductions 
of  the  models. 


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Lessons  in  Physical  Geography 

By  CHARLES  R.  DRYER,  M.A.,  F.G.S.A. 
Professor  of  Geography  in  the  Indiana  State  Normal  School 


Half  leather,  12mo.     Illustrated.     430  pages.      .       .       ,       Price,  $1.20 


EASY  AS  WELL  AS  FULL  AND  ACCURATE 

One  of  the  chief  merits  of  this  text-book  is  that  it  is  simpler  than 
any  other  complete  and  accurate  treatise  on  the  subject  now  before  the 
public.  The  treatment,  although  specially  adapted  for  the  high  school 
course,  is  easily  within  the  comprehension  of  pupils  in  the  upper  grade 
of  the  grammar  school. 

TREATMENT  BY  TYPE  FORMS 

The  physical  features  of  the  earth  are  grouped  according  to  their 
causal  relations  and  their  functions.  The  characteristics  of  each  group 
are  presented  by  means  of  a  typical  example  which  is  described  in  unusual 
detail,  so  that  the  pupil  has  a  relatively  minute  knowledge  of  the  type  form. 

INDUCTIVE  GENERALIZATIONS 

Only  after  the  detailed  discussion  of  a  t5rpe  form  has  given  the  pupil 
a  clear  and  vivid  concept  of  that  form  are  explanations  and  general  prin- 
ciples introduced.  Generalizations  developed  thus  inductively  rest  upon 
an  adequate  foundation  in  the  mind  of  the  pupil,  and  hence  cannot 
appear  to  him  mere  formulae  of  words,  as  is  too  often  the  case. 

REALISTIC  EXERCISES 

Throughout  the  book  are  many  realistic  exercises  which  include  both 
field  and  laboratory  work.  In  the  field,  the  student  is  taught  to  observe 
those  physiographic  forces  which  may  be  acting,  even  on  a  small  scale, 
in  his  own  immediate  vicinity.  Appendices  (with  illustrations)  give  full 
instructions  as  to  laboratory  material  and  appliances  for  observation  and 
for  teaching. 

SPECIAL  ATTENTION  TO  SUBJECTS  OF  HUMAN   INTEREST 

While  due  prominence  is  given  to  recent  developments  in  the  study, 
this  does  not  exclude  any  link  in  the  chain  which  connects  the  face  of  the 
earth  with  man.  The  chapters  upon  life  contain  a  fuller  and  more 
adequate  treatment  of  the  controls  exerted  by  geographical  conditions 
upon  plants,  animals,  and  man  than  has  been  given  in  any  other  similar 
book. 

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and  reproductions  of  photographs,  but  illustrations  have  been  used  only 
\yhere  they  afford  real  aid  in  the  elucidation  of  the  text. 


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